Disjoint Hamiltonian Cycles in Bipartite Graphs

Size: px
Start display at page:

Download "Disjoint Hamiltonian Cycles in Bipartite Graphs"

Transcription

1 Disjoint Hamiltonian Cycles in Bipartite Graphs Michael Ferrara 1, Ronald Gould 1, Gerard Tansey 1 Thor Whalen Abstract Let G = (X, Y ) be a bipartite graph and define σ (G) = min{d(x) + d(y) : xy / E(G), x X, y Y }. Moon and Moser [6] showed that if G is a bipartite graph on n vertices such that σ (G) n + 1 then G is hamiltonian, extending a classical result of Ore [7]. Here we prove that if G is a bipartite graph on n vertices such that σ (G) n + k 1 then G contains k edge-disjoint hamiltonain cycles. This extends the result of Moon and Moser and a result of R. Faudree, et al. [3] Keywords: Graph, Degree Sum, Bipartite, Disjoint Hamiltonain Cycles. 1 Introduction and Terminology For any graph G, let V (G) and E(G) V (G) V (G) denote the sets of vertices and edges of G respectively. An edge between two vertices x and y in V (G) shall be denoted xy. Furthermore, let δ(g) denote the minimum degree of a vertex in G. A useful reference for any undefined terms is [1]. If G is bipartite with bipartition (X, Y ) we will write G = (X, Y ). If X = Y then we will say that G is balanced. A proper pair in G is a pair of non-adjacent vertices (x, y) with x in X and y in Y. A cycle in G is a sequence of distinct vertices v 1, v 1,..., v t such that v i v i+1 (1 i t 1) and v 1 v t are in E(G). We shall denote a cycle on t vertices by C t. A hamiltonian cycle in a graph G is a cycle that spans V (G) and, if such a cycle exists, G is said to be hamiltonian. Hamiltonian graphs and their properties have been widely studied. A good reference for recent developments and open problems is [5]. In general, we are interested in degree conditions that assure hamiltonian cycles in a graph. Perhaps the most classical condition of this type was put forth by Dirac []. Theorem 1.1 (Dirac 195). If G be a graph on n 3 vertices such that δ(g) n, then G is hamiltonian. 1 Department of Mathematics and Computer Science, Emory University, Atlanta, GA Metron Inc., Reston, VA 1

2 For an arbitrary graph G, define σ (G) to be the minimum degree sum of non-adjacent vertices in G. Of greater interest for our work here is the famed Ore condition for hamiltonicity which uses this parameter. Theorem 1. (Ore 1960). If G is a graph of order n 3 such that σ (G) n, then G is hamiltonian. The reader should note that Dirac s theorem is a corollary of Ore s theorem. In a bipartite graph G, we are interested instead in the parameter σ (G), defined to be the minimum degree sum of a proper pair. Moon and Moser [6] extended Ore s theorem to bipartite graphs as follows. Theorem 1.3 (Moon, Moser 1960). If G = (X, Y ) is a balanced bipartite graph on n vertices such that σ (G) n + 1, then G is hamiltonian. Faudree, Rousseau and Schelp [3] were able to give degree-sum conditions that assured the existence of many disjoint hamiltonian cycles in an arbitrary graph. Theorem 1.4 (Faudree, Rousseau, Schelp 1984). If G is a graph on n vertices such that σ (G) n + k then for n sufficiently large, G contains k edge-disjoint hamiltonian cycles. In this paper we will extend the previous two results by proving the following. Theorem 1.5. If G = (X, Y ) is a balanced bipartite graph of order n, with n 18k such σ (G) n + k 1, then G has k edge-disjoint hamiltonian cycles. Veneering Numbers and k-extendibility To prove our main theorem, we need some results on path-systems in bipartite graphs. Our strategy is to develop k systems of edge-disjoint paths and show that they can be extended to k edge-disjoint hamiltonian cycles. The following definitions and theorems can be found in [4]. Let W k (G) be the family of all k-sets {(w 1, z 1 ),..., (w k, z k )} of pairs of vertices of G where w 1,..., w k, z 1,..., z k are all distinct. Let S k (G) denote the collection of edge-disjoint path systems in G that have exactly k paths. If W W k (G) lists the end-points of a path system P in S k (G), we say that P is a W -linkage. A graph G is said to be k-linked if there is a W -linkage for every W W k (G). A graph G is said to be k-extendible if any W -linkage of maximal order is spanning. In order to tailor the idea of extendible path systems to bipartite graphs, we introduce the idea of a veneering path system. Definition 1. A path system P veneers a bipartite graph G if it covers all the vertices of one of the partite sets.

3 Given a W W k (G), we denote by W 1 the set of bipartite pairs of W, by W X those pairs of W that are in X, and by W Y those that are in Y. Also, with a slight abuse of notion, we will let W X (resp. W Y ) be the set of vertices of X (resp. Y ) that are used in the pairs of W. Definition. Let G be a bipartite graph and W W k (G). The veneering number ϑ X Y (W ) of W is defined to be ϑ X Y (W ) = (X Y ) ( W X W Y ), = (X Y ) W X W Y. For a given path system P, let (P) denote the set of pairs of endpoints of paths in P and let P denote P (P). We define the veneering number of such a P to be the veneering number of P. The veneering number of a given set of endpoints is of interest, because it represents the minimum possible number of vertices left uncovered by a path system with those endpoints. We can now reformulate the notion of a k-extendible graph. If P 1 and P are two path-systems of G, we write P 1 P when every path of P 1 is contained in a path of P. The following fact will prove most useful. Proposition.1. Let G = (X Y, E) be a bipartite graph and P 1, P S(G) be such that P 1 P. Let G 1 = (X 1 Y 1, E 1 ) = G P 1, G = (X Y, E ) = G P, then ϑ X 1 Y 1 (P 1 ) = ϑ X Y (P ). We are now ready to give our definition of a k-extendible bipartite graph. Definition 3. Let G be a bipartite graph. Then G is said to be k-extendible if for any path-system P in S k (G) there exists some veneering path system P in S k (G) that preserves the endpoints of P. We will utilize the following in the proof of our main theorem. Theorem.. If k and G = (X, Y ) is a bipartite graph of order n such that X, Y > 3k and σ n+3k (G), then G is k-extendible. It is important to note that a maximal path system with veneering number zero is spanning. Thus, if a graph G that meets the σ bound for k-extendibility has some path system P in S k (G) such that ϑ(p) = 0, then G must have a spanning path system. We give two more results from [4] that will be very useful. 3

4 Theorem.3. If G = (X Y, E) is a balanced bipartite graph of order n with σ (G) n + k then for any set W in W k (G) comprised entirely of proper pairs of G, there exists a system of k edge-disjoint paths whose endpoints are exactly the pairs in W. Theorem.4. If G = (X Y, E) is a balanced bipartite graph of order n such that for any x X and any y Y, d(x) + d(y) n +, then for any pair (x, y) of vertices of G, there exists a Hamilton path between x and y. The degree sum condition is the best possible. 3 Main Theorem Before we begin the proof of Theorem 1.5, we need to establish a number of facts. Suppose the theorem is not true, and let G be a counterexample of order n with a maximum number of edges. The maximality of G implies that for any proper pair (x, y), G+xy contains k edge-disjoint Hamilton cycles, one of these containing the edge xy. Thus, to a proper pair (x, y) we will associate k 1 edge-disjoint Hamilton cycles H 1,..., H k 1 and an (x, y)-hamilton path P = (x = z 1, z,, z n = y). Let H denote the union of subgraphs H 1,..., H k 1, and L = L(x, y) denote the subgraph obtained from G by removing the edges of H. Before we go on proving our theorem we will state a few facts about L. Throughout these proofs, we must keep in mind that n 18k, (1) and for any vertex w of G, we have d L (w) = d G (w) (k 1), () so the degree sum condition on any proper pair (x, y) of G d G (x) + d G (y) n + k 1. (3) This yields the following: Fact 1. For any proper pair (x, y) of G, we have d L (x) + d L (y) n k + 3. (4) Fact. If there is a proper pair (x, y) of G, with v d G (x) + d G (y) n + 4k 3, or equivalently d L (x) + d L (y) n + 1, then L contains a Hamilton cycle. 4

5 If there were a proper pair (x, y) of G such that d G (x) + d G (y) n + 4k 3, then by (), d L (x) + d L (y) n + 1, hence if we consider the (x, y)-path P in L, we see that there must be a vertex z V (P ) such that z N(y) and z + N(x). Then xz + [z +, y] P yz [x, z] is a hamiltonian cycle in L. Note that the existence of P shows that L is connected In fact, L must be -connected. Lemma 3.1. If L is not -connected, then there are k edge-disjoint hamiltonian cycles in G. Proof: Suppose w is a cut-vertex of L; we assume, without loss of generality, that w X. Since L admits a h1amiltonian path, L w can only have two components, one of them being balanced. Let B be the subgraph of G induced by the balanced component of L w and A = G B. Note that w A, and E L (A X w, B) = E L (A Y, B) =. Let a = A X = A Y and b = B X = B Y. Claim 1. a, b > n k. Proof: Assume a n k. Then a(k ) + a < ak n, implying a(k ) < n a = b, so d H (A Y, B X ) < B X = b. Thus there is a vertex u B X such that E H (u, A Y ) =, so E G (u, A Y ) =. Take any v A Y. We have uv / E(G), so d(u) + d(v) A X + d H (v, B X ) + B Y + d H (u, A Y ) a + (k 1) + b < n + k 1, which contradicts the condition of our theorem. Claim 1 The two following claims give lower bounds on the degrees of the vertices in L. Claim. For any z A w, d L (z) A k+3 and for any z B, d L(z) B k+3. Proof: Assume z B Y (the cases z B X, z A X, z A Y are similar). By Claim 1 and the fact that n 18k, we have A X w = a 1 > n k 1 > (k 1), so there is a z A X w such that zz / E(H), thus zz / E(G), so that d L (z) + d L (z ) n k + 3. Then since d L (z ) a, we get d L (z) n k + 3 a = b k + 3. Claim Claim 3. d L (w) n k k + 3. Proof: If w is adjacent, in G, to all the vertices of A Y, then the theorem is obviously true. If not, there is a v A Y with wv / E(G), so that d L (w) + d L (v) n k + 3. Since d L (v) a = n b < n n k, we get d L (w) n k + 3 d L (v) > n k + 3 (n n k ) = n k + 3. k 5

6 Claim 3 Finally: Claim 4. E G (A X, B Y ), E G (A Y, B X ) k 1. Proof: If G[(A X, B Y )] is complete, the result is obvious. If not, there is a pair of nonadjacent vertices u A X and v B Y, so d(u) + d(v) n + k 1. Yet d(u, A Y ) a and d(v, B X ) b, so d(u, B Y ) + d(v, A X ) n + k 1 a b = k 1. The proof is identical for (A Y, B X ). Claim 4 By Claim, Claim 3, and the fact that n 18k we have, for any pair of vertices (u, v) V (A X ) V (A Y ) d A (u) + d A (v) A k n k k + 3 > A + k. Thus, A, and by a similar computation B, satisfies the conditions of Theorem.4. Hence take k pairs (e i, e i ) of edges such that the e i are distinct edges of E G (A X, B Y ) and the e i are distinct edges of E G (A Y, B X ). These edges exist by Claim 4. Let u i A X and v i B Y be the end vertices of e i, and u i A X and v i B Y be the end vertices of e i. Since A and B both satisfy the conditions of Theorem.4, there are k edge-disjoint hamiltonian paths U 1,, U k in A such that u i and u i are the end-vertices of U i, and there are k edge-disjoint hamiltonian paths V 1,, V k in B such that v i and v i are the end-vertices of V i. Together with the e i and e i edges we get k edges disjoint hamiltonian cycles in G, which contradicts the assumption. Hence the lemma is proven. Now we show that the -connectedness of L assures the existence of a large cycle. Lemma 3.. If L is -connected, then it contains a cycle of order at least n 4k + 4. Proof: Let x 1 X and y n Y be two non-adjacent vertices of G and let P = x 1 y 1 x n y n be a hamiltonian path in L. This path induces an obvious ordering of the vertices; namely, we say that z < z if one encounters z before z when traversing P from x 1 to y n. We say that a vertex z is the minimum (maximum) vertex with respect to a given property if z < z (z < z) for all other vertices z satisfying that property. By the -connectivity of L, N(x 1 ) and N(y n ) are non-empty. Let x be the minimum vertex of N(y n ) and y be the maximum vertex of N(x 1 ). Note that if x < y, then the path along C between x and y has length at least 4k 1, since otherwise the xy-path must have length at most 4k 3, and C = x 1 y [y, y n ] y n x [x, x 1 ] would form a cycle of length at least n 4k + 4. But then d L (y n ) n 1 d L (x 1 ) k, 6

7 which gives that d L (x 1 )+d L (y n ) n k 1, a contradiction of the degree-sum condition. So y < x. We construct a cycle C using the following algorithm: Set z 0 = x 1 and z 3 = y Set i = 0 While z i+3 / [x +, y n ] do Set i = i + 1 Let z i and z i+3 be vertices of P such that z i < z i+1 < z i+3, z i z i+3 E(L), and z i+3 is maximum (note that the existence of this edge is ensured by the -connectivity of L) End While Set l = i + Set z 1 = x 1 and replace z 3 with the minimum vertex of N(x 1 ) [z +, y] Set z l = y n and z l be the maximum vertex of N(y n ) [x, z l 1 ] Set C = z 1 z 3 z l z l l i=1 z iz i+3 l i=1 [z i 1, z i ] Note that since y < x, the while loop is performed at least once, and l 3. We now show that C n 4k + 4. Let F = {x i y i (x i / N(y n )) (y i / N(x 1 ))}. Note that there cannot be any i with x i N(y n ) and y i N(x 1 ), since then [x 1, x i ] [y i, y n ] x 1 y i x i y n would be a hamiltonian cycle. This implies that F + {x i y i : y i N(x 1 )} + {x i y i : x i N(y n )} = F + d L (x 1 ) + d L (y n ) = n By Fact 1 we get that F k 3 (5) We show that all but perhaps vertices of V (G C ) are in V (F ). All the vertices of [z +, z 3 ], except z 3, are in V (F ) by the minimality of z 3. Similarly, by the maximality of z l, the only vertex that is in V (F ) [z + l, z l 1 ] is z+ l. Finally, by the maximality of y and the minimality of x we get V ( l 1 i= [z+ i, z i+1 ]) V ([y+, x ]) V (F ), hence (L V (F )) V (L C ). By (5) we get L C F + 4k 4, hence C n 4k

8 3.1 Path Systems In order to prove an important technical lemma, we must first establish some facts about extending paths and path systems. Lemma 3.3. Let G = (X Y, E) be a bipartite graph, and let P be a path system of G. Let X be a subset of ( P) X, and let Y be a subset of Y ( P) Y. Suppose that X = s + t, where s is the number of vertices in X arising from paths of P consisting of a single vertex. If δ(x, Y ) > t + s then there exists another path system, P, of G such that P P and ( P ) X =. Proof:We will first show that s may be assumed to be 0. If s > 0, let P 1, P,..., P s be the trivial paths contained in X. Now, for every i [s], replace P i = {x i } with a path P i on three vertices such that the endvertices of P i are new vertices added to X and the middle vertex of P i is a new vertex added to Y Y. In addition, let the endvertices of P i be adjacent to the neighbors of x i. Let P 1 be the new path system, and let X 1, consisting of X and the vertices added to X, be the new set of endvertices we wish to eliminate. The new system P 1 now contains no trivial paths, and X 1 = t + s. Thus, if our lemma were true for systems with no trivial paths, then the condition δ(x, Y ) > t + s = t + s ensures the existence of a path system P 1 such that P 1 P 1 and ( P 1 ) X 1 =. By replacing every P i by P i within the appropriate paths of P 1, we obtain the desired path system of G. So assume that X = {x 1,..., x t }. Note that the result clearly holds if t = 1, so assume that t. We exhibit an algorithm that produces a sequence of path systems P = P(0) P(1) P(t) = P where P(i) is obtained from P(i 1) by adding an edge from x i to Y, and P is the desired path system. Since this algorithm will attach an edge to every vertex of X, P will have no endvertices in X. Given a vertex x i X, let P i P(i 1) be the path having x i as an endvertex, and let w i be the other endvertex of P i. Let Z i be the set of internal vertices of the paths of P(i 1) in Y. The algorithm consists of the following steps: 8

9 1) Set P(0) = P ) For i = 1 to t do 3) Let y be any vertex of N(x i, Y ) w i Z i 4) Set P(i) to be P(i 1) with the path P i replaced with P i x i y 5) end of ) loop Clearly this algorithm will terminate if N(x i, Y ) w i Z i is never empty when the vertex y must be chosen in step 3), and if the algorithm terminates, we will have the desired path system. Note that since no path in P(0) had internal vertices in Y, the only way that a vertex a Y can be an internal vertex of a path of P(i) is if the algorithm selected a in step 3) on two passes through the algorithm. Thus which implies that Z i max{0, i } t, N(x i, Y ) w i Z i d(x, Y ) 1 Z i > t 1 t = 0. The following corollary is obtained from Lemma 3.3 by induction on k: Corollary 3.4. Let G = (X Y, E) be a bipartite graph, let P 1,..., P k be k edge-disjoint path systems, and let Y Y k i=1 int(p i) Y. For all i [k] let X i ( P i ) X and X i = s i + t i, where s i is the number of vertices of X i arising from paths of P i consisting of a single vertex. If for all i [k], δ(x i, Y ) > t i + s i + (k 1) then there exist k edge-disjoint path systems P 1,..., P k such that for all i [k], P i P i and ( P i ) X i =. 3. The Degree-Product Lemma Interestingly, the proof of Theorem 1.5 relies on a result pertaining to degree products as opposed to degree sums. We feel it would be interesting to investigate similar results. Lemma 3.5. If G has no proper pair (u, v) such that d L (u)d L (v) 1k(n 1k) then G has k edge-disjoint hamiltonian cycles. Proof: Suppose G has no such vertices. Let A be the subgraph of G generated by the vertices of degree less than 16k, and B the subgraph generated by the vertices of degree greater or equal to 16k. By (3) and (1) no bipartite pairs (u, v) of A are proper. Further, 9

10 no bipartite pairs (u, v) of B can be proper or else, by (), (1), Fact 1 and convexity, we would have d L (u)d L (v) (14k + )(n 16k + 1) 1k(n 1k). Thus A and B induce complete bipartite graphs. Assume without loss of generality, that A X A Y, and set λ = A X A Y = B Y B X. We can assume λ < 4k 3 since otherwise we could find a proper non-adjacent pair (x, y) V (B X ) V (A Y ) with d G (x) + d G (y) B Y + λ + A X + λ = n + λ n + 4k 3, and Fact would imply a Hamilton cycle in L, hence k edge-disjoint Hamilton cycles in G. Claim 5. We have δ(a X, B Y ) λ + k 1 and δ(a Y, B X ) k 1 λ Let x A X such that d(x, B Y ) = δ(a X, B Y ). By (3), every vertex y B Y N(x, B Y ) must verify so d G (y) n + k 1 d G (x) = n + k 1 A Y d(x, B Y ) = B Y + k 1 δ(a X, B Y ), d G (y, A X ) B Y + k 1 δ(a X, B Y ) B X = λ + k 1 δ(a X, B Y ) This implies that d G (A X x, B Y N(x, B Y )) ( B Y δ(a X, B Y ))(λ + k 1 δ(a X, B Y )) (6) yet, since the vertices of A X can be adjacent to no more than λ + 4k 1 vertices of B Y (by Fact ), we see that d G (A X x, B Y N(x, B Y )) ( A X 1)(λ + 4k 1). (7) Thus if λ + k 1 δ(a X, B Y ) > 0, (6) and (7) imply B Y ( A X 1)(λ + 4k 1) λ + k 1 δ(a X, B Y ) + δ(a X, B Y ) (16k)(8k 4) + k which contradicts the fact that n 18k, hence δ(a X, B Y ) λ + k 1. The proof of δ(a Y, B X ) k 1 λ is similar. Claim5 We distinguish two cases, according to the size of A X : 10

11 Case 1: Suppose 1 A Y k 1. Then Claim 5 and the completeness of A imply δ(a Y ) A X + k 1 λ = A Y + k 1 > A Y + (k 1). Now, we apply Corollary 3.4 with P i = X i = A Y for all i, and let Y = X. This implies, in the language of the corollary, that δ(x i, Y ) = δ(a Y ). Thus, we find that there are k edge-disjoint systems P 1,..., P k whose paths have all order three and whose endvertices are all in X. Further, since A is a complete bipartite graph, we may choose these path-systems so that they cover a subset A X of min( A X, A Y ) vertices of A X. That is to say, if A X A Y, A X = A X, so these systems each cover A entirely, and if A X > A Y, we require that they each cover the same proper subset A X of A X having order A Y. For all i [k] we let P i = P i when A X A Y, and P i = P i (A X A X ) when A X > A Y. In either case, we now have k edge-disjoint path systems which cover A. Again we wish to apply Corollary 3.4 to the P i with X i = ( P i ) A X, to extend to a family of k edge-disjoint systems P 1,..., P k such that every path in each of these systems has both endvertices in B. We may do so since if A X A Y then all t i = A X vertices of X i come from non-trivial paths, and if A X > A Y then t i = A Y vertices of X i also come from non-trivial paths, and s i = A X A Y of them come from paths consisting of exactly one vertex, so by Claim 5, d(a X, B Y ) λ + k 1 > t i + s i + (k 1). Consider some matching M 1 that contains exactly one edge from each non-empty path in P 1. Clearly, ϑx Y (M 1) = 0, and therefore by Proposition.1 we have that ϑ( (P 1)) = 0 (8) in G P 1. Thus, as (P 1 ) B, and B induces a complete bipartite graph, we can link the endpoints of the paths in P 1 to form a Hamiltonian cycle in G. Suppose then that we have extended P 1,..., P t 1 (t k) to the disjoint Hamiltonian cycles H 1,..., H t 1. As above, Proposition.1 implies that ϑ( (P t)) = 0 (9) in G P t. Assume that P t has exactly j paths, and let {x 1, y 1 },..., {x j, y j } denote the pairs of endpoints of these paths. Additionally, let the set W = {{y 1, x }, {y, x 3 },..., {y j, x 1 }}. As B induces a complete bipartite graph with each partite set having size at least n 11

12 A Y λ n 6k, it is simple to see that there is a W -linkage in G t := G P t t 1 i=1 E(H i). Note that there are at most j A Y < k paths in P t, so if we are able to show that G t is k-extendible we will be done. By Corollary., it suffices to show that σ (G t ) > V (G t) + 6k n k n k. (10) In G, the minimum degree of a vertex in the subgraph induced by B is n ( A Y + λ) n 6k. In removing the edges from the t 1 other hamiltonian cycles, each vertex loses t < k adjacencies. Thus, it is clear that σ (G t) certainly exceeds n k, completing this case. Case : Suppose A Y k. Let A X be a subset of A Y vertices of A X. As A is a complete bipartite graph, there are k edge-disjoint hamiltonian cycles in (A X A Y ) G, and we let x 1 y 1,, x k y k be independent edges of (A X A Y ) G such that x i y i is an edge of the i th hamiltonian cycle. Using Claim 5 we get that δ(a X, B Y ) k 1 and δ(a Y, B X ) k 1 λ so Let B = G A X A Y. We have δ(a Y, B X) A X A X + δ(a Y, B X ) k 1. σ (B ) δ(a X A X, B Y ) + B X B X + λ + k 1 = B + k 1 One may then use the edges of E(A X, B Y ) and E(A Y, X A X ) along with Theorem.3 to find k edge-disjoint hamiltonian cycles in G. Before we proceed to prove the main theorem, we give one final technical lemma. Lemma 3.6. Let G be a graph containing a hamiltonian cycle C and let S and R be nonempty disjoint subsets of V (G). If R + S E(R, S) then there are four distinct vertices c 1, c, c 3, c 4, encountered in that order on C, such that either (a) c 1, c 3 R, c, c 4 S, c 1 c E(G), and c 3 c 4 E(G), or (b) c 1, c 4 S, c, c 3 R, c 1 c 3 E(G), and c c 4 E(G). (c) c 1, c 4 R, c, c 3 S, c 1 c 3 E(G), and c c 4 E(G). Proof: First, note that if R = {r R : d(r, S) = 0} and S = {s S : d(s, R) = 0}, then R + S R + S E(R, S) = E(R, S ) 1

13 so we may assume that every vertex of R is adjacent to at least one vertex of S, and vica versa. Further, observe that the inequality in the statement of the lemma cannot hold if R = 1 or S = 1, so R, S. Let R = {u 1,..., u r }, where the labels of the vertices of R are determined by a chosen orientation of C. Let C and C be the two [u 1, u r ]-paths of C, C is the path containing the u i for 1 i r. In order to avoid (a), (b), and (c), N(R) S must be entirely contained in C and for every 1 i < j r, N(u i, S) N(u j, S) must be empty if i j +, and can have at most one element if i = j + 1. Thus, to avoid (a), (b) and (c), S must contain at least E(R, S) R + 1 E(R, S) R + 1 vertices. 3.3 Proof of Theorem 1.5 Proof: Let C be a cycle of L of maximal order which minimizes d L (T, C), where T = L C. By Lemma 3. t = T k (11) Let u T X and v T Y such that d L (u, C) + d L (v, C) is maximal. Let α = d L (u, C) and β = d L (v, C). We assume, without loss of generality, that α β. We may assume that α k + 4. (1) Indeed, by Fact 1, every vertex of V (Y ) N(u) has degree greater or equal to n k + 3 t α. If α k+3, this would yield that there are at least n t (k+3) (k 1) n 6k vertices that have degree at least n k + 3 t (k + 3) n 6k, implying that G is hamiltonian by Lemma 3.5. Note that or else C could be extended. α + β n t + 1 n k + 3 We must have N L (u, C) + N L (v, C) 1 and N L (u, C) N L (v, C) + 1. Let R = N L (v, C) + N G (u, C). Then R d L (v, C) d H (u, C) N L (u, C) N L (v, C) + β (k 1) 1 = β k + 1 (13) For every r R ru / E(G), so by Fact 1, d L (r) + d L (u) = d L (r, T ) + d L (r, C) + d L (u) n k + 3, 13

14 hence d L (r, C) n k + 3 d L (u, C) d L (u, T ) d L (r, T ) (14) n k + 3 α t t Together with the fact that r R d L(r, T ) t 1 (since otherwise, we could extend C), we get d L (R, C) = r R d L (r, C) r R ( n k + 3 dl (u, C) d L (u, T ) d L (r, T ) ) (15) = R (n k + 3) R (α + t) r R d L (r, T ) (16) R (n k + 3 α t) t + 1. (17) Let S = N L (u, C). We have d L (R, S) d L (R, C) C X S R (n k + 3 α t) t + 1 (n t) + S = R (n k + 3 α t) + S + 1 n Lemma 3.6 with R = R and S = S + gives S ( d L (R, S) R + 1 ) 0 S ( ( R (n k + 3 α t) + S + 1 n) R + 1 ) 0 n R (n k + α t) 0 (18) By (13) and (1), we have R α k + 1 3, so (18) yields n 3(n k + α t) 0 (19) 3α n k + 9 (0) α 3 n k 3t + 3. (1) 3 Yet, as α β, t k 1, and n 18k 46k), this would imply α + β 4 3 n 4 k 6(k 1) + 6 > n + k 3 contradicting (3.3). T heorem

15 References [1] G. Chartrand, L. Lesniak, Graphs and Digraphs 3 rd Edition, CRC Press, [] G.A. Dirac, Some Theorems on Abstract Graphs, Proc. London Math. Soc., (195), [3] R. Faudree, C. Rousseau and R. Schelp, Edge-Disjoint Hamiltonian Cycles, Graph Theory with Applications to Algorithms and COmputer Science, 1984, [4] R. Gould and T. Whalen, Connectivity, Linkage and Hamilton Path-Systems in Bipartite Graphs.(preprint) [5] R. Gould, Advances on the Hamiltonian Problem, A Survey, Graphs and Combinatorics, 19 (003), 7-5. [6] J. Moon and L. Moser, On Hamiltonian Bipartite Graphs, Israel J. Math, 1 (1963), MR 8 # 4540 [7] O. Ore, A Note on Hamilton Circuits., Amer. Math. Monthly, 67 (1960),

Discrete Mathematics

Discrete Mathematics Discrete Mathematics 309 (009) 3811 380 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Disjoint hamiltonian cycles in bipartite graphs Michael

More information

Distance between two k-sets and Path-Systems Extendibility

Distance between two k-sets and Path-Systems Extendibility Distance between two k-sets and Path-Systems Extendibility December 2, 2003 Ronald J. Gould (Emory University), Thor C. Whalen (Metron, Inc.) Abstract Given a simple graph G on n vertices, let σ 2 (G)

More information

HAMILTONIAN CYCLES AVOIDING SETS OF EDGES IN A GRAPH

HAMILTONIAN CYCLES AVOIDING SETS OF EDGES IN A GRAPH HAMILTONIAN CYCLES AVOIDING SETS OF EDGES IN A GRAPH MICHAEL J. FERRARA, MICHAEL S. JACOBSON UNIVERSITY OF COLORADO DENVER DENVER, CO 8017 ANGELA HARRIS UNIVERSITY OF WISCONSIN-WHITEWATER WHITEWATER, WI

More information

Graphs and Combinatorics

Graphs and Combinatorics Graphs and Combinatorics (007) 3:165 18 Digital Object Identifier (DOI) 10.1007/s00373-006-0665-0 Graphs and Combinatorics Springer-Verlag 007 Subdivision Extendibility Ronald Gould 1 and Thor Whalen 1

More information

Observation 4.1 G has a proper separation of order 0 if and only if G is disconnected.

Observation 4.1 G has a proper separation of order 0 if and only if G is disconnected. 4 Connectivity 2-connectivity Separation: A separation of G of order k is a pair of subgraphs (H, K) with H K = G and E(H K) = and V (H) V (K) = k. Such a separation is proper if V (H) \ V (K) and V (K)

More information

Observation 4.1 G has a proper separation of order 0 if and only if G is disconnected.

Observation 4.1 G has a proper separation of order 0 if and only if G is disconnected. 4 Connectivity 2-connectivity Separation: A separation of G of order k is a pair of subgraphs (H 1, H 2 ) so that H 1 H 2 = G E(H 1 ) E(H 2 ) = V (H 1 ) V (H 2 ) = k Such a separation is proper if V (H

More information

List of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2,

List of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2, List of Theorems Mat 416, Introduction to Graph Theory 1. Ramsey s Theorem for graphs 8.3.11. Theorem 1 The numbers R(p, q) exist and for p, q 2, R(p, q) R(p 1, q) + R(p, q 1). If both summands on the

More information

CYCLE STRUCTURES IN GRAPHS. Angela K. Harris. A thesis submitted to the. University of Colorado Denver. in partial fulfillment

CYCLE STRUCTURES IN GRAPHS. Angela K. Harris. A thesis submitted to the. University of Colorado Denver. in partial fulfillment CYCLE STRUCTURES IN GRAPHS by Angela K. Harris Master of Science, University of South Alabama, 003 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for

More information

THE STRUCTURE AND EXISTENCE OF 2-FACTORS IN ITERATED LINE GRAPHS

THE STRUCTURE AND EXISTENCE OF 2-FACTORS IN ITERATED LINE GRAPHS Discussiones Mathematicae Graph Theory 27 (2007) 507 526 THE STRUCTURE AND EXISTENCE OF 2-FACTORS IN ITERATED LINE GRAPHS Michael Ferrara Department of Theoretical and Applied Mathematics The University

More information

An Ore-type Condition for Cyclability

An Ore-type Condition for Cyclability Europ. J. Combinatorics (2001) 22, 953 960 doi:10.1006/eujc.2001.0517 Available online at http://www.idealibrary.com on An Ore-type Condition for Cyclability YAOJUN CHEN, YUNQING ZHANG AND KEMIN ZHANG

More information

be a path in L G ; we can associated to P the following alternating sequence of vertices and edges in G:

be a path in L G ; we can associated to P the following alternating sequence of vertices and edges in G: 1. The line graph of a graph. Let G = (V, E) be a graph with E. The line graph of G is the graph L G such that V (L G ) = E and E(L G ) = {ef : e, f E : e and f are adjacent}. Examples 1.1. (1) If G is

More information

arxiv: v1 [cs.dm] 12 Jun 2016

arxiv: v1 [cs.dm] 12 Jun 2016 A Simple Extension of Dirac s Theorem on Hamiltonicity Yasemin Büyükçolak a,, Didem Gözüpek b, Sibel Özkana, Mordechai Shalom c,d,1 a Department of Mathematics, Gebze Technical University, Kocaeli, Turkey

More information

Graphs with large maximum degree containing no odd cycles of a given length

Graphs with large maximum degree containing no odd cycles of a given length Graphs with large maximum degree containing no odd cycles of a given length Paul Balister Béla Bollobás Oliver Riordan Richard H. Schelp October 7, 2002 Abstract Let us write f(n, ; C 2k+1 ) for the maximal

More information

4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**

4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu** 4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published

More information

Antoni Marczyk A NOTE ON ARBITRARILY VERTEX DECOMPOSABLE GRAPHS

Antoni Marczyk A NOTE ON ARBITRARILY VERTEX DECOMPOSABLE GRAPHS Opuscula Mathematica Vol. 6 No. 1 006 Antoni Marczyk A NOTE ON ARBITRARILY VERTEX DECOMPOSABLE GRAPHS Abstract. A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n

More information

Ring Sums, Bridges and Fundamental Sets

Ring Sums, Bridges and Fundamental Sets 1 Ring Sums Definition 1 Given two graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) we define the ring sum G 1 G 2 = (V 1 V 2, (E 1 E 2 ) (E 1 E 2 )) with isolated points dropped. So an edge is in G 1 G

More information

On the Turán number of forests

On the Turán number of forests On the Turán number of forests Bernard Lidický Hong Liu Cory Palmer April 13, 01 Abstract The Turán number of a graph H, ex(n, H, is the maximum number of edges in a graph on n vertices which does not

More information

On a Conjecture of Thomassen

On a Conjecture of Thomassen On a Conjecture of Thomassen Michelle Delcourt Department of Mathematics University of Illinois Urbana, Illinois 61801, U.S.A. delcour2@illinois.edu Asaf Ferber Department of Mathematics Yale University,

More information

arxiv: v1 [math.co] 28 Oct 2016

arxiv: v1 [math.co] 28 Oct 2016 More on foxes arxiv:1610.09093v1 [math.co] 8 Oct 016 Matthias Kriesell Abstract Jens M. Schmidt An edge in a k-connected graph G is called k-contractible if the graph G/e obtained from G by contracting

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

More information

ON GLOBAL DOMINATING-χ-COLORING OF GRAPHS

ON GLOBAL DOMINATING-χ-COLORING OF GRAPHS - TAMKANG JOURNAL OF MATHEMATICS Volume 48, Number 2, 149-157, June 2017 doi:10.5556/j.tkjm.48.2017.2295 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst

More information

Set-orderedness as a generalization of k-orderedness and cyclability

Set-orderedness as a generalization of k-orderedness and cyclability Set-orderedness as a generalization of k-orderedness and cyclability Keishi Ishii Kenta Ozeki National Institute of Informatics, Tokyo 101-8430, Japan e-mail: ozeki@nii.ac.jp Kiyoshi Yoshimoto Department

More information

Improved degree conditions for 2-factors with k cycles in hamiltonian graphs

Improved degree conditions for 2-factors with k cycles in hamiltonian graphs Improved degree conditions for -factors with k cycles in hamiltonian graphs Louis DeBiasio 1,4, Michael Ferrara,, Timothy Morris, December 4, 01 Abstract In this paper, we consider conditions that ensure

More information

Extensions of a theorem of Erdős on nonhamiltonian graphs

Extensions of a theorem of Erdős on nonhamiltonian graphs Extensions of a theorem of Erdős on nonhamiltonian graphs Zoltán Füredi Alexandr Kostochka Ruth Luo March 9, 017 Abstract arxiv:1703.1068v [math.co] 5 Apr 017 Let n, d be integers with 1 d, and set hn,

More information

Discrete Mathematics. The edge spectrum of the saturation number for small paths

Discrete Mathematics. The edge spectrum of the saturation number for small paths Discrete Mathematics 31 (01) 68 689 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The edge spectrum of the saturation number for

More information

Combined degree and connectivity conditions for H-linked graphs

Combined degree and connectivity conditions for H-linked graphs Combined degree and connectivity conditions for H-linked graphs FLORIAN PFENDER Universität Rostock Institut für Mathematik D-18055 Rostock, Germany Florian.Pfender@uni-rostock.de Abstract For a given

More information

HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS. 1. Introduction

HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS. 1. Introduction HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS VALERY S. GORDON, YURY L. ORLOVICH, FRANK WERNER Abstract. A triangular grid graph is a finite induced subgraph of the infinite graph associated with the

More information

and critical partial Latin squares.

and critical partial Latin squares. Nowhere-zero 4-flows, simultaneous edge-colorings, and critical partial Latin squares Rong Luo Department of Mathematical Sciences Middle Tennessee State University Murfreesboro, TN 37132, U.S.A luor@math.wvu.edu

More information

Decompositions of graphs into cycles with chords

Decompositions of graphs into cycles with chords Decompositions of graphs into cycles with chords Paul Balister Hao Li Richard Schelp May 22, 2017 In memory of Dick Schelp, who passed away shortly after the submission of this paper Abstract We show that

More information

Saturation numbers for Ramsey-minimal graphs

Saturation numbers for Ramsey-minimal graphs Saturation numbers for Ramsey-minimal graphs Martin Rolek and Zi-Xia Song Department of Mathematics University of Central Florida Orlando, FL 3816 August 17, 017 Abstract Given graphs H 1,..., H t, a graph

More information

Partitioning a graph into highly connected subgraphs

Partitioning a graph into highly connected subgraphs Partitioning a graph into highly connected subgraphs Valentin Borozan 1,5, Michael Ferrara, Shinya Fujita 3 Michitaka Furuya 4, Yannis Manoussakis 5, Narayanan N 5,6 and Derrick Stolee 7 Abstract Given

More information

SATURATION SPECTRUM OF PATHS AND STARS

SATURATION SPECTRUM OF PATHS AND STARS 1 Discussiones Mathematicae Graph Theory xx (xxxx 1 9 3 SATURATION SPECTRUM OF PATHS AND STARS 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 Jill Faudree Department of Mathematics and Statistics University of

More information

Graph Theory. Thomas Bloom. February 6, 2015

Graph Theory. Thomas Bloom. February 6, 2015 Graph Theory Thomas Bloom February 6, 2015 1 Lecture 1 Introduction A graph (for the purposes of these lectures) is a finite set of vertices, some of which are connected by a single edge. Most importantly,

More information

AALBORG UNIVERSITY. Total domination in partitioned graphs. Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo

AALBORG UNIVERSITY. Total domination in partitioned graphs. Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo AALBORG UNIVERSITY Total domination in partitioned graphs by Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo R-2007-08 February 2007 Department of Mathematical Sciences Aalborg University Fredrik

More information

Group Colorability of Graphs

Group Colorability of Graphs Group Colorability of Graphs Hong-Jian Lai, Xiankun Zhang Department of Mathematics West Virginia University, Morgantown, WV26505 July 10, 2004 Abstract Let G = (V, E) be a graph and A a non-trivial Abelian

More information

HAMILTONICITY AND FORBIDDEN SUBGRAPHS IN 4-CONNECTED GRAPHS

HAMILTONICITY AND FORBIDDEN SUBGRAPHS IN 4-CONNECTED GRAPHS HAMILTONICITY AND FORBIDDEN SUBGRAPHS IN 4-CONNECTED GRAPHS FLORIAN PFENDER Abstract. Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3, 3, 3,,,,, ), i.e. T is a chain

More information

Even Cycles in Hypergraphs.

Even Cycles in Hypergraphs. Even Cycles in Hypergraphs. Alexandr Kostochka Jacques Verstraëte Abstract A cycle in a hypergraph A is an alternating cyclic sequence A 0, v 0, A 1, v 1,..., A k 1, v k 1, A 0 of distinct edges A i and

More information

Partitioning a graph into highly connected subgraphs

Partitioning a graph into highly connected subgraphs Partitioning a graph into highly connected subgraphs Valentin Borozan 1,5, Michael Ferrara, Shinya Fujita 3 Michitaka Furuya 4, Yannis Manoussakis 5, Narayanan N 6 and Derrick Stolee 7 Abstract Given k

More information

How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected?

How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected? How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected? Michael Anastos and Alan Frieze February 1, 2018 Abstract In this paper we study the randomly

More information

Automorphism groups of wreath product digraphs

Automorphism groups of wreath product digraphs Automorphism groups of wreath product digraphs Edward Dobson Department of Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State, MS 39762 USA dobson@math.msstate.edu Joy

More information

arxiv: v2 [math.co] 7 Jan 2016

arxiv: v2 [math.co] 7 Jan 2016 Global Cycle Properties in Locally Isometric Graphs arxiv:1506.03310v2 [math.co] 7 Jan 2016 Adam Borchert, Skylar Nicol, Ortrud R. Oellermann Department of Mathematics and Statistics University of Winnipeg,

More information

Highly Hamiltonian Graphs and Digraphs

Highly Hamiltonian Graphs and Digraphs Western Michigan University ScholarWorks at WMU Dissertations Graduate College 6-017 Highly Hamiltonian Graphs and Digraphs Zhenming Bi Western Michigan University, zhenmingbi@gmailcom Follow this and

More information

Paths and cycles in extended and decomposable digraphs

Paths and cycles in extended and decomposable digraphs Paths and cycles in extended and decomposable digraphs Jørgen Bang-Jensen Gregory Gutin Department of Mathematics and Computer Science Odense University, Denmark Abstract We consider digraphs called extended

More information

EXCLUDING SUBDIVISIONS OF INFINITE CLIQUES. Neil Robertson* Department of Mathematics Ohio State University 231 W. 18th Ave. Columbus, Ohio 43210, USA

EXCLUDING SUBDIVISIONS OF INFINITE CLIQUES. Neil Robertson* Department of Mathematics Ohio State University 231 W. 18th Ave. Columbus, Ohio 43210, USA EXCLUDING SUBDIVISIONS OF INFINITE CLIQUES Neil Robertson* Department of Mathematics Ohio State University 231 W. 18th Ave. Columbus, Ohio 43210, USA P. D. Seymour Bellcore 445 South St. Morristown, New

More information

Some Nordhaus-Gaddum-type Results

Some Nordhaus-Gaddum-type Results Some Nordhaus-Gaddum-type Results Wayne Goddard Department of Mathematics Massachusetts Institute of Technology Cambridge, USA Michael A. Henning Department of Mathematics University of Natal Pietermaritzburg,

More information

Hamilton Cycles in Digraphs of Unitary Matrices

Hamilton Cycles in Digraphs of Unitary Matrices Hamilton Cycles in Digraphs of Unitary Matrices G. Gutin A. Rafiey S. Severini A. Yeo Abstract A set S V is called an q + -set (q -set, respectively) if S has at least two vertices and, for every u S,

More information

All Ramsey numbers for brooms in graphs

All Ramsey numbers for brooms in graphs All Ramsey numbers for brooms in graphs Pei Yu Department of Mathematics Tongji University Shanghai, China yupeizjy@16.com Yusheng Li Department of Mathematics Tongji University Shanghai, China li yusheng@tongji.edu.cn

More information

K 4 -free graphs with no odd holes

K 4 -free graphs with no odd holes K 4 -free graphs with no odd holes Maria Chudnovsky 1 Columbia University, New York NY 10027 Neil Robertson 2 Ohio State University, Columbus, Ohio 43210 Paul Seymour 3 Princeton University, Princeton

More information

Out-colourings of Digraphs

Out-colourings of Digraphs Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.

More information

Long Cycles in 3-Connected Graphs

Long Cycles in 3-Connected Graphs Long Cycles in 3-Connected Graphs Guantao Chen Department of Mathematics & Statistics Georgia State University Atlanta, GA 30303 Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta,

More information

Discrete Mathematics. The average degree of a multigraph critical with respect to edge or total choosability

Discrete Mathematics. The average degree of a multigraph critical with respect to edge or total choosability Discrete Mathematics 310 (010 1167 1171 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The average degree of a multigraph critical with respect

More information

On saturation games. May 11, Abstract

On saturation games. May 11, Abstract On saturation games Dan Hefetz Michael Krivelevich Alon Naor Miloš Stojaković May 11, 015 Abstract A graph G = (V, E) is said to be saturated with respect to a monotone increasing graph property P, if

More information

SEMI-STRONG SPLIT DOMINATION IN GRAPHS. Communicated by Mehdi Alaeiyan. 1. Introduction

SEMI-STRONG SPLIT DOMINATION IN GRAPHS. Communicated by Mehdi Alaeiyan. 1. Introduction Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 2 (2014), pp. 51-63. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir SEMI-STRONG SPLIT DOMINATION

More information

1.3 Vertex Degrees. Vertex Degree for Undirected Graphs: Let G be an undirected. Vertex Degree for Digraphs: Let D be a digraph and y V (D).

1.3 Vertex Degrees. Vertex Degree for Undirected Graphs: Let G be an undirected. Vertex Degree for Digraphs: Let D be a digraph and y V (D). 1.3. VERTEX DEGREES 11 1.3 Vertex Degrees Vertex Degree for Undirected Graphs: Let G be an undirected graph and x V (G). The degree d G (x) of x in G: the number of edges incident with x, each loop counting

More information

On Hamiltonian cycles and Hamiltonian paths

On Hamiltonian cycles and Hamiltonian paths Information Processing Letters 94 (2005) 37 41 www.elsevier.com/locate/ipl On Hamiltonian cycles and Hamiltonian paths M. Sohel Rahman a,, M. Kaykobad a,b a Department of Computer Science and Engineering,

More information

Cycles with consecutive odd lengths

Cycles with consecutive odd lengths Cycles with consecutive odd lengths arxiv:1410.0430v1 [math.co] 2 Oct 2014 Jie Ma Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. Abstract It is proved that there

More information

EQUITABLE COLORING OF SPARSE PLANAR GRAPHS

EQUITABLE COLORING OF SPARSE PLANAR GRAPHS EQUITABLE COLORING OF SPARSE PLANAR GRAPHS RONG LUO, D. CHRISTOPHER STEPHENS, AND GEXIN YU Abstract. A proper vertex coloring of a graph G is equitable if the sizes of color classes differ by at most one.

More information

Decomposing planar cubic graphs

Decomposing planar cubic graphs Decomposing planar cubic graphs Arthur Hoffmann-Ostenhof Tomáš Kaiser Kenta Ozeki Abstract The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree,

More information

Compatible Circuit Decompositions of 4-Regular Graphs

Compatible Circuit Decompositions of 4-Regular Graphs Compatible Circuit Decompositions of 4-Regular Graphs Herbert Fleischner, François Genest and Bill Jackson Abstract A transition system T of an Eulerian graph G is a family of partitions of the edges incident

More information

Independent Transversals in r-partite Graphs

Independent Transversals in r-partite Graphs Independent Transversals in r-partite Graphs Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract Let G(r, n) denote

More information

Parity Versions of 2-Connectedness

Parity Versions of 2-Connectedness Parity Versions of 2-Connectedness C. Little Institute of Fundamental Sciences Massey University Palmerston North, New Zealand c.little@massey.ac.nz A. Vince Department of Mathematics University of Florida

More information

On the connectivity of the direct product of graphs

On the connectivity of the direct product of graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 41 (2008), Pages 45 56 On the connectivity of the direct product of graphs Boštjan Brešar University of Maribor, FEECS Smetanova 17, 2000 Maribor Slovenia bostjan.bresar@uni-mb.si

More information

The Turán number of sparse spanning graphs

The Turán number of sparse spanning graphs The Turán number of sparse spanning graphs Noga Alon Raphael Yuster Abstract For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic

More information

On colorability of graphs with forbidden minors along paths and circuits

On colorability of graphs with forbidden minors along paths and circuits On colorability of graphs with forbidden minors along paths and circuits Elad Horev horevel@cs.bgu.ac.il Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva 84105, Israel Abstract.

More information

Minimal Paths and Cycles in Set Systems

Minimal Paths and Cycles in Set Systems Minimal Paths and Cycles in Set Systems Dhruv Mubayi Jacques Verstraëte July 9, 006 Abstract A minimal k-cycle is a family of sets A 0,..., A k 1 for which A i A j if and only if i = j or i and j are consecutive

More information

Tough graphs and hamiltonian circuits

Tough graphs and hamiltonian circuits Discrete Mathematics 306 (2006) 910 917 www.elsevier.com/locate/disc Tough graphs and hamiltonian circuits V. Chvátal Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada Abstract

More information

Partial cubes: structures, characterizations, and constructions

Partial cubes: structures, characterizations, and constructions Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes

More information

Hamilton cycles and closed trails in iterated line graphs

Hamilton cycles and closed trails in iterated line graphs Hamilton cycles and closed trails in iterated line graphs Paul A. Catlin, Department of Mathematics Wayne State University, Detroit MI 48202 USA Iqbalunnisa, Ramanujan Institute University of Madras, Madras

More information

Spanning Paths in Infinite Planar Graphs

Spanning Paths in Infinite Planar Graphs Spanning Paths in Infinite Planar Graphs Nathaniel Dean AT&T, ROOM 2C-415 600 MOUNTAIN AVENUE MURRAY HILL, NEW JERSEY 07974-0636, USA Robin Thomas* Xingxing Yu SCHOOL OF MATHEMATICS GEORGIA INSTITUTE OF

More information

Extremal H-colorings of trees and 2-connected graphs

Extremal H-colorings of trees and 2-connected graphs Extremal H-colorings of trees and 2-connected graphs John Engbers David Galvin June 17, 2015 Abstract For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the

More information

A Generalization of a result of Catlin: 2-factors in line graphs

A Generalization of a result of Catlin: 2-factors in line graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 72(2) (2018), Pages 164 184 A Generalization of a result of Catlin: 2-factors in line graphs Ronald J. Gould Emory University Atlanta, Georgia U.S.A. rg@mathcs.emory.edu

More information

Decomposing oriented graphs into transitive tournaments

Decomposing oriented graphs into transitive tournaments Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote

More information

Multipartite tournaments with small number of cycles

Multipartite tournaments with small number of cycles Multipartite tournaments with small number of cycles Gregory Gutin and Arash Rafiey Department of Computer Science Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK Gutin(Arash)@cs.rhul.ac.uk

More information

Compatible Circuit Decompositions of Eulerian Graphs

Compatible Circuit Decompositions of Eulerian Graphs Compatible Circuit Decompositions of Eulerian Graphs Herbert Fleischner, François Genest and Bill Jackson Septemeber 5, 2006 1 Introduction Let G = (V, E) be an Eulerian graph. Given a bipartition (X,

More information

1 Hamiltonian properties

1 Hamiltonian properties 1 Hamiltonian properties 1.1 Hamiltonian Cycles Last time we saw this generalization of Dirac s result, which we shall prove now. Proposition 1 (Ore 60). For a graph G with nonadjacent vertices u and v

More information

Degree Conditions for Spanning Brooms

Degree Conditions for Spanning Brooms Degree Conditions for Spanning Brooms Guantao Chen, Michael Ferrara, Zhiquan Hu, Michael Jacobson, and Huiqing Liu December 4, 01 Abstract A broom is a tree obtained by subdividing one edge of the star

More information

Packing and decomposition of graphs with trees

Packing and decomposition of graphs with trees Packing and decomposition of graphs with trees Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel. e-mail: raphy@math.tau.ac.il Abstract Let H be a tree on h 2 vertices.

More information

Edge Disjoint Cycles Through Specified Vertices

Edge Disjoint Cycles Through Specified Vertices Edge Disjoint Cycles Through Specified Vertices Luis Goddyn Ladislav Stacho Department of Mathematics Simon Fraser University, Burnaby, BC, Canada goddyn@math.sfu.ca, lstacho@math.sfu.ca February 16, 2005

More information

Hamilton-Connected Indices of Graphs

Hamilton-Connected Indices of Graphs Hamilton-Connected Indices of Graphs Zhi-Hong Chen, Hong-Jian Lai, Liming Xiong, Huiya Yan and Mingquan Zhan Abstract Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald

More information

ON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2. Bert L. Hartnell

ON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2. Bert L. Hartnell Discussiones Mathematicae Graph Theory 24 (2004 ) 389 402 ON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2 Bert L. Hartnell Saint Mary s University Halifax, Nova Scotia, Canada B3H 3C3 e-mail: bert.hartnell@smu.ca

More information

Compatible Hamilton cycles in Dirac graphs

Compatible Hamilton cycles in Dirac graphs Compatible Hamilton cycles in Dirac graphs Michael Krivelevich Choongbum Lee Benny Sudakov Abstract A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated

More information

Longest paths in strong spanning oriented subgraphs of strong semicomplete multipartite digraphs

Longest paths in strong spanning oriented subgraphs of strong semicomplete multipartite digraphs Longest paths in strong spanning oriented subgraphs of strong semicomplete multipartite digraphs Gregory Gutin Department of Mathematical Sciences Brunel, The University of West London Uxbridge, Middlesex,

More information

M INIM UM DEGREE AND F-FACTORS IN GRAPHS

M INIM UM DEGREE AND F-FACTORS IN GRAPHS NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 33-40 M INIM UM DEGREE AND F-FACTORS IN GRAPHS P. K a t e r in is a n d N. T s ik o p o u l o s (Received June 1998) Abstract. Let G be a graph, a,

More information

Small Cycle Cover of 2-Connected Cubic Graphs

Small Cycle Cover of 2-Connected Cubic Graphs . Small Cycle Cover of 2-Connected Cubic Graphs Hong-Jian Lai and Xiangwen Li 1 Department of Mathematics West Virginia University, Morgantown WV 26505 Abstract Every 2-connected simple cubic graph of

More information

arxiv: v1 [cs.dm] 26 Apr 2010

arxiv: v1 [cs.dm] 26 Apr 2010 A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs George B. Mertzios Derek G. Corneil arxiv:1004.4560v1 [cs.dm] 26 Apr 2010 Abstract Given a graph G, the longest path

More information

arxiv: v1 [math.co] 13 May 2016

arxiv: v1 [math.co] 13 May 2016 GENERALISED RAMSEY NUMBERS FOR TWO SETS OF CYCLES MIKAEL HANSSON arxiv:1605.04301v1 [math.co] 13 May 2016 Abstract. We determine several generalised Ramsey numbers for two sets Γ 1 and Γ 2 of cycles, in

More information

Domination and Total Domination Contraction Numbers of Graphs

Domination and Total Domination Contraction Numbers of Graphs Domination and Total Domination Contraction Numbers of Graphs Jia Huang Jun-Ming Xu Department of Mathematics University of Science and Technology of China Hefei, Anhui, 230026, China Abstract In this

More information

Extremal H-colorings of graphs with fixed minimum degree

Extremal H-colorings of graphs with fixed minimum degree Extremal H-colorings of graphs with fixed minimum degree John Engbers July 18, 2014 Abstract For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices

More information

An approximate version of Hadwiger s conjecture for claw-free graphs

An approximate version of Hadwiger s conjecture for claw-free graphs An approximate version of Hadwiger s conjecture for claw-free graphs Maria Chudnovsky Columbia University, New York, NY 10027, USA and Alexandra Ovetsky Fradkin Princeton University, Princeton, NJ 08544,

More information

Upper Bounds of Dynamic Chromatic Number

Upper Bounds of Dynamic Chromatic Number Upper Bounds of Dynamic Chromatic Number Hong-Jian Lai, Bruce Montgomery and Hoifung Poon Department of Mathematics West Virginia University, Morgantown, WV 26506-6310 June 22, 2000 Abstract A proper vertex

More information

Average degrees of edge-chromatic critical graphs

Average degrees of edge-chromatic critical graphs Average degrees of edge-chromatic critical graphs Yan Cao a,guantao Chen a, Suyun Jiang b,c, Huiqing Liu d, Fuliang Lu e a Department of Mathematics and Statistics, Georgia State University, Atlanta, GA

More information

Rao s degree sequence conjecture

Rao s degree sequence conjecture Rao s degree sequence conjecture Maria Chudnovsky 1 Columbia University, New York, NY 10027 Paul Seymour 2 Princeton University, Princeton, NJ 08544 July 31, 2009; revised December 10, 2013 1 Supported

More information

MINIMALLY NON-PFAFFIAN GRAPHS

MINIMALLY NON-PFAFFIAN GRAPHS MINIMALLY NON-PFAFFIAN GRAPHS SERGUEI NORINE AND ROBIN THOMAS Abstract. We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-pfaffian graphs minimal with respect

More information

On the Pósa-Seymour Conjecture

On the Pósa-Seymour Conjecture On the Pósa-Seymour Conjecture János Komlós, 1 Gábor N. Sárközy, 2 and Endre Szemerédi 3 1 DEPT. OF MATHEMATICS, RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903 2 DEPT. OF COMPUTER SCIENCE, WORCESTER POLYTECHNIC

More information

Solution to a problem on hamiltonicity of graphs under Oreand Fan-type heavy subgraph conditions

Solution to a problem on hamiltonicity of graphs under Oreand Fan-type heavy subgraph conditions Noname manuscript No. (will be inserted by the editor) Solution to a problem on hamiltonicity of graphs under Oreand Fan-type heavy subgraph conditions Bo Ning Shenggui Zhang Binlong Li Received: date

More information

Fractional and circular 1-defective colorings of outerplanar graphs

Fractional and circular 1-defective colorings of outerplanar graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 6() (05), Pages Fractional and circular -defective colorings of outerplanar graphs Zuzana Farkasová Roman Soták Institute of Mathematics Faculty of Science,

More information

Rainbow Matchings of Size δ(g) in Properly Edge-Colored Graphs

Rainbow Matchings of Size δ(g) in Properly Edge-Colored Graphs Rainbow Matchings of Size δ(g) in Properly Edge-Colored Graphs Jennifer Diemunsch Michael Ferrara, Allan Lo, Casey Moffatt, Florian Pfender, and Paul S. Wenger Abstract A rainbow matching in an edge-colored

More information

The Algorithmic Aspects of the Regularity Lemma

The Algorithmic Aspects of the Regularity Lemma The Algorithmic Aspects of the Regularity Lemma N. Alon R. A. Duke H. Lefmann V. Rödl R. Yuster Abstract The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in

More information

On two conjectures about the proper connection number of graphs

On two conjectures about the proper connection number of graphs On two conjectures about the proper connection number of graphs Fei Huang, Xueliang Li, Zhongmei Qin Center for Combinatorics and LPMC arxiv:1602.07163v3 [math.co] 28 Mar 2016 Nankai University, Tianjin

More information

Zero-Sum Flows in Regular Graphs

Zero-Sum Flows in Regular Graphs Zero-Sum Flows in Regular Graphs S. Akbari,5, A. Daemi 2, O. Hatami, A. Javanmard 3, A. Mehrabian 4 Department of Mathematical Sciences Sharif University of Technology Tehran, Iran 2 Department of Mathematics

More information