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

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1 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 a hamiltonian graph has a -factor with exactly k cycles. Brandt et al. proved that if G is a graph on n 4k vertices with minimum degree at least n, then G contains a -factor with exactly k cycles; moreover this is best possible. Faudree et al. asked if there is some c < 1 such that δ(g) cn would imply the existence of a -factor with k-cycles under the additional hypothesis that G was hamiltonian. This question was answered in the affirmative by Sárközy, who used the regularity blow-up method to show that there exists some ε > 0 such that for every k 1 and large n, δ(g) ( 1 ε) n suffices. We improve on this result, giving an elementary proof that for every ε > 0 and k 1, if G is a hamiltonian graph on n k ε vertices with δ(g) ( + ε) n, then G contains a -factor with k cycles. Keywords: -factor, hamiltonian cycle 1 Introduction A -factor of a graph G is a spanning -regular subgraph or, alternatively is a collection of disjoint cycles that spans G. As a hamiltonian cycle is a -factor with precisely one component, the problem of determining when a graph G has a -factor is a natural and well-studied relaxation of the hamiltonian problem. A number of results concerning the existence of -factors, along with many related cycle and factorization problems, can be found in [1, 8, 9, 1]. It has been demonstrated a number of times that established sufficient conditions for the existence of some cycle-structural property can be relaxed if the graphs under consideration are also assumed to be hamiltonian. For instance, in [] Amar et al. proved that the minimum degree threshold for pancyclicity in general graphs from [] can be significantly lowered in (necessarily nonbipartite) hamiltonian graphs. 1 Department of Mathematics, Miami University; debiasld@miamioh.edu Dept. of Mathematical and Statistical Sciences, Univ. of Colorado Denver; {michael.ferrara, timothy.morris}@ucdenver.edu Research supported in part by Simons Foundation Collaboration Grant # Research supported in part by Simons Foundation Collaboration Grant #407. 1

2 Theorem 1. If G is a hamiltonian graph of order n and and δ(g) n+1, then G is either pancyclic or bipartite. In [4], Brandt et al. proved the following. Theorem. If k 1 is an integer and G is a graph of order n 4k such that δ(g) n, then G contains a -factor with exactly k cycles. In fact, it was shown that the weaker Ore-type degree condition σ (G) n suffices in place of the given assumption on the minimum degree. Motivated by Theorems 1 and, Faudree et al. proved the following result in [6]. Theorem. If G is a hamiltonian graph of order n 6 and δ(g) 1n +, then G contains a -factor with two components. In [14], Pfender showed that under the additional assumption that G is claw-free, the minimum degree threshold in Theorem can be greatly reduced to δ(g) 7, which is sharp. Theorem inspired the following conjecture, which also appeared in [6]. Conjecture 1. For each integer k there exists a positive real number c k < 1 and integers a k and n k such that every hamiltonian graph G of order n > n k with δ(g) c k n + a k contains a -factor with k cycles. Conjecture 1 was affirmed by Sárközy [17] using the regularity blow-up method. Theorem 4. There exists a real number ε > 0 such that for every integer k there exists an integer n 0 = n 0 (k) such that every hamiltonian graph G of order n n 0 with δ(g) (1/ ε)n has a -factor with k components. Sárközy s result raises a natural question: what is the minimum δ = δ(k, n) such that every hamiltonian graph of order n with minimum degree at least δ has a -factor with exactly k cycles. In 01, Győri and Li announced [10] that they have shown δ ( 11 + ε) n suffices for n sufficiently large. Our main result is the following improvement of Theorem, Theorem 4, and that result. Theorem. Let k 1 and 0 < ε < If G is a hamiltonian graph on n k ε vertices with δ(g) ( + ε) n, then G contains a -factor with exactly k cycles. Additionally, our proof uses only elementary techniques, and hence requires a significantly smaller bound on n than that required by Theorem 4. One vexing aspect of Conjecture 1 and the related work described here is that it is possible that a sublinear, or even constant, minimum degree would suffice to ensure a hamiltonian graph has a -factor of the desired type. Specifically, while the 4-regular graph obtained by replacing every vertex of a cycle with a copy of K e is an example of a family of hamiltonian graphs that do not contain a -factor with two cycles, currently we know of no example that has δ.

3 Proof of Theorem A key ingredient in our proof is the following lemma of Nash-Williams [1]. For completeness, we provide a brief sketch of the proof using modern terminology. Lemma 1. Let G be a -connected graph on n vertices. If δ(g) =: d n+, then either G has a hamiltonian cycle or α(g) d + 1. Proof (sketch). Choose a longest cycle C in G, which has length at least min{δ(g), G } by a classical result of Dirac []. If some component of G C is a vertex v, then the successors of the neighbors of v on C along with v form an independent set of size at least d + 1. Otherwise, let P be a longest path in G C and suppose that P has length at least 1. Consider the endpoints of P say x 1 and x p ; all of their neighbors are in V (P ) V (C). If v i V (C) is a neighbor of x 1, then x p cannot be adjacent to any of the p vertices preceding v i or the p vertices succeeding v i. Counting edges between {x 1, x p } and C in two ways gives a contradiction. We also need the following well known result of Pósa [16]. Theorem 6. Let G be a graph on n vertices and for all i [n] let S i = {v V (G) : deg(v) i}. If for all 1 i < n 1, S i < i and S (n 1)/ n 1, then G has a hamiltonian cycle. Finally, a convex geometric graph G is a graph whose vertices are embedded in the plane in convex position, and whose edges (subject to the embedding of V (G)) are straight line segments. We will utilize the following extremal result of Kupitz [11] (see [7], Theorem 1.9). Theorem 7. Let G be a convex geometric graph of order n k + 1. If E(G) > (k 1)n, then G contains k pairwise noncrossing edges. We are now ready to prove Theorem. Proof. Let k, 0 < ε < 1 10, and n k ε. Note that this implies k 1 εn. We consider two cases, based on the connectivity of G. Case 1: κ(g) < εn. Let S be a minimum cut-set of G. As G is hamiltonian, we have S < εn and further, since δ(g S) ( + ε) n S > n, G S has exactly two components. Let X and Y be those two components, and suppose without loss of generality that X Y. Note then that n < Y n X < n. (1) Consequently, each pair of adjacent vertices u and v in Y satisfies (N(u) N(v)) Y 4 n Y 4 ( Y ) Y = Y. So we may remove a family T of k vertex-disjoint triangles from Y and set Y = Y \ T T V (T ). Now, let X = {v S : deg(v, X) > εn}, Y = S \ X, G 1 = G[X X ], and G = G[Y Y ]. We now verify that G 1 and G satisfy the conditions of Theorem 6, in which case the hamiltonian cycles in G 1 and G along with T give us the desired -factor with k cycles.

4 First note that by (1), X X n so Y Y n. Now for all y Y, ( ) ( ) deg(y, G ) + ε n deg(y, X) S Y \ Y ε n n > Y, and for all y Y, deg(y, G ) ( ) + ε n X n ( Y Y ) 4 Y Y. These calculations show that S i = for all i < n and S i < n i for all n i Y Y. Likewise, by (1), Y > n and thus X X < n. Now for all x X, and for all x X, deg(x, G 1 ) deg(x, G 1 ) > εn > X ( ) + ε n Y n ( ) X X = X X. These calculations show that S i = for all i < εn and S i < εn i for all εn i X X. Case : κ(g) εn. Suppose first that α(g) < n, and let x 1,..., x k 1 be distinct vertices in V (G). Since ( + ε) n (k 1) > n > α(g), we can iteratively construct a family of k 1 vertex disjoint triangles each consisting of x i and some edge in N(x i ). Let G be the graph obtained by deleting these k 1 triangles. As ( ) δ(g ) + ε n (k 1) > n > α(g) α(g ), and κ(g ), Lemma 1 implies that G is hamiltonian, completing the desired -factor with k cycles. Therefore, we may assume that α(g) n. Let H be a hamiltonian cycle in G and let X = {x 1, x,..., x t } be a largest independent set in G. Assuming that H has an implicit clockwise orientation, for each i [t], let y i be the successor of x i along H and let Y = {y 1, y,..., y t }. Note that since X is independent, X Y =. Let D be an auxiliary directed graph with vertex set V (D) = {(x 1, y 1 ),..., (x t, y t )} such that ((x i, y i ), (x j, y j )) E(D) if and only if x i y j is an edge in G (see (a) and (b) in Figure 1). As each vertex v in G has at least δ(g) (n X Y ) ( + ε) n (n t) neighbors in X Y, and n t, we have that δ + (D) ( + ε ) ( ) ( ) ( ) 1 n (n t) = t ε n t ε t = + ε t. () Thus e(d) ( 1 + ε) t and consequently the number of -cycles in D is at least εt. Finally, let Γ be a convex geometric graph with V (Γ) = V (D) such that {(x i, y i ), (x j, y j )} E(Γ) if and only if {(x i, y i ), (x j, y j )} induces a -cycle in D. By (), e(γ) εt, so Theorem 7 implies that Γ contains at least εt pairwise disjoint edges. These εt εn k pairwise disjoint edges correspond to non-overlapping pairs of consecutive chords on H (as in part (a) of Figure 1), which allow us to split H into a -factor with k cycles, completing the proof 4

5 (a) The hamiltonian cycle H with vertices from X colored white and vertices from Y colored black (b) The auxiliary directed graph D Figure 1 (c) The auxiliary convex geometric graph Γ References [1] J. Akiyama and M. Kano, Factors and Factorizations of Graphs: Proof Techniques in Factor Theory, Lecture Notes in Mathematics Vol. 01, Springer, 011. [] D. Amar, E. Flandrin, I. Fournier and A. Germa, Pancyclism in Hamiltonian Graphs, Discrete Math. 89 (1991), [] J. A. Bondy, Pancyclic graphs I, J. Combin. Theory, Ser. B 11 (1971), [4] S. Brandt, G. Chen, R. Faudree, R. Gould and L. Lesniak, Degree Conditions for -Factors, J. Graph Theory 4 (1997), [] G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (19), [6] R. Faudree, R. Gould, M. Jacobson, L. Lesniak and A. Saito, A Note on -factors with Two Components, Discrete Math. 00 (00), [7] S. Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Advanced Lectures in Mathematics, Springer, 004. [8] R. Gould, Advances on the Hamiltonian Problem A Survey, Graphs and Combinatorics 19 (00), 7. [9] R. Gould, A look at cycles containing specified elements of a graph, Discrete Math. 09 (009), [10] E. Gyori and H. Li, -factors in Hamiltonian graphs, in preparation. [11] Y. Kupitz, On pairs of disjoint segments in convex position in the plane, Annals Discr. Math. 0 (1984), 0 08.

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