Research Prospectus. Murong Xu Department of Mathematics, West Virginia University, Morgantown, WV 26506

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1 Research Prospectus Murong Xu Department of Mathematics, West Virginia University, Morgantown, WV The Problems Motivating The Research This research project is motivated by coloring problems in graphs and connectivity problems in digraphs. We consider finite simple graphs and digraphs without loops but parallel arcs. Undefined terms and notations will follow [2] for graphs and [1] for digraphs. Through this prospectus, G denotes an (undirected) graph with vertex set V (G) and edge set E(G); D denotes a digraph with vertex set V (D) and arc set A(D); (G) and δ(g) denote the maximum degree and the minimum degree of a graph G, respectively. 1.1 Motivation of Research on Coloring Problems For integers k, r 1, a k-coloring of a graph G is a mapping from V (G) to the set of colors k = {1, 2,..., k} such that any two adjacent vertices receive different colors. A (k, r)-coloring of a graph G is a proper k-coloring c : V (G) k such that for every vertex v, the neighborhood of v contains at least min{d(v), r} different colors. The r-hued chromatic number of G, denoted by χ r (G), is the smallest k such that G has a (k, r)-coloring. If r = 2, a (k, r)-coloring of G is also called a dynamic k-coloring. For a positive integer h, denote 2 h to be the power set of h. Let L : V (G) 2 h be an assignment with L(v) being a list of available colors at v. An L-coloring is a proper coloring c such that c(v) L(v), for every v V (G). When such c exists, we say that G is L-colorable. For any v V (G), if L(v) = k, then L is a k-element list. For a given assignment L of G, an (L, r)-coloring of G is an L-coloring as well as an r-hued coloring. The list r-hued chromatic number of G, denoted by χ L,r (G), is the least k such that for every k-element list L, G has a (L, r)-coloring. The concept of linear coloring was introduced by Yuster [35]. A linear k-coloring of a graph G is a proper coloring of G such that the subgraph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number χ l (G) of G is the smallest k such that G has a linear k-coloring. A (k, r)-coloring c of a graph G is linear if c is also a linear coloring. The linear r-hued chromatic number of G, denoted by χ l r(g), is the smallest k such 1

2 that G admits a linear (k, r)-coloring. The linear list chromatic number of G, denoted by χ l L (G), is the smallest k such that G admits a linear L-coloring for every k-element list L. For a given assignment L of G, a linear (L, r)-coloring c of G is an L-coloring as well as a linear r-hued coloring. The linear list r-hued chromatic number of G, denoted by χ l L,r (G), is the smallest k such that for every k-element list L, G always admits a linear (L, r)-coloring. There have been many fascinating conjectures related to graph coloring problems. Among them are the following. Wegner in [33] posed the following conjecture. Conjecture 1.1. (Wegner [33]) If G is a planar graph, then { (G) + 5, if 4 (G) 7; χ (G) 3 (G)/2 + 1, if (G) 8. For each integer r 1, define r + 3, if 1 r 2; f 1 (r) = r + 5, if 3 r 7; 3r/2 + 1, if r 8. Motivated by Wegner s conjecture, Song et al. raised a more general one on the upper bounds of χ r (G) when G is a planar graph. Conjecture 1.2. (Song et al. [30]) Let G be a planar graph. Then we have χ r (G) f 1 (r). Thus Wegner s conjecture is a special case when r =. A graph G is K 4 -minor free if G does not have a subgraph contractible to K 4. Define { r + 3, if 2 r 3; K(r) = 3r/2 + 1, if r 4. Lih et al. proved the following towards Wegner s conjecture. Theorem 1.1. (K.-W. Lih, W.-F. Wang and X. Zhu [22]). Let G be a K 4 -minor free graph. Then χ (G) K( (G)). Song et al. extended Theorem 1.1 to the following theorem. Theorem 1.2. (H. Song et al. [30]) Let G be a K 4 -minor free graph with = (G), and r 2 be an integer. Then (i) χ r (G) K(r). (ii) χ L,r (G) K(r) + 1. Problem 1. Can we extend Theorem 1.2 to linear r-hued coloring? 2

3 When r = 1, Conjecture 1.2 becomes the 4-color-theorem. It is of interest to study the case when r = 2 for Conjecture 1.2. In [7], Chen et al. proved that χ 2 (G) 5 for any planar graph G and conjectured that χ 2 (G) 4 if G is a connected planar graph other than C 5. Kim et al. [16] proved this conjecture by using Four Coloring Theorem. Theorem 1.3. (Kuratowski s Theorem) A graph is planar if and only if it contains no subdivision of either K 5 or K 3,3. By Kuratowski s Theorem, it is natural to consider the upper bound of 2-hued chromatic numbers of graphs having no K 5 minors or no K 3,3 minors. For a connected graph G without K 5 minor, Kim et al. proved the following theorem. Theorem 1.4. (Y.-J. Kim, S.-J. Lee and S.-I. Oum [17]) If G is a connected graph other than C 5 having no K 5 minor, the G is dynamically 4-colorable. Problem 2. What is the upper bound of chromatic number of dynamic k-coloring of K 3,3 -minor free graphs? The maximum subgraph average degree of G, denoted by Mad(G), is defined by Mad(G) = max{ 2 E(H) V (H), H G}. It is noted that if G is a planar graph, then Mad(G) < 6. Therefore, it is natural to investigate χ r (G) and χ l L,r (G) for graphs G with bounded values of Mad(G). Song et al. in [31] proved the following theorem. Theorem 1.5. (H. Song et al. [31]) If r 3 and G is a planar graph with g(g) 6, then χ r (G) r + 5. Problem 3. For bounded values of Mad(G), can we find the upper bounds of χ l L,r (G) of a graph G? 1.2 Motivation of Research on Digraph Connectivity Problems For a graph G, let κ (G) and τ(g) denote edge-connectivity and the number of edge-disjoint spanning trees of G, respectively. For a digraph D, λ(d) denotes the arc-strong connectivity of D. For any disjoint subsets X, Y V (D), define(x, Y ) D to be the set of all arcs in D with tail in X and head in Y, and let + D (X) = (X, V (D) X) D and D (X) = (V (D) X, X) D, and let d + D (X) = + D (X) and d D (X) = D (X). When X = {v}, we write d+ D (v) = + D ({v}) (called the out degree of v in D) and d D (v) = D ({v}) (called the in degree of v in D). A digraph D is k-arc-strong if for any nonempty proper subset X V (D), d + D (X) k. Thus λ(d) is the largest integer k such that D is k-arc-strong. Catlin et al. in [3] proved a characterization of κ (G) in terms of τ(g). 3

4 Theorem 1.6. (Catlin et al., Theorem 1.1 of [3]) Let G be a connected graph and let k 1 be an integer. Each of the following holds. (i) κ (G) 2k if and only if X E(G) with X k, τ(g X ) k. (ii) κ (G) 2k + 1 if and only if X E(G) with X k + 1, τ(g X ) k. An arborescence is an oriented tree T such that for some r V (T ), we have d T (r) = 0, and for any v V (T ) r, we have d T (v) = 1. The vertex r is the root of T and T is an r-arborescence. Let τ(d) be the maximum number of arc-disjoint spanning arborescences in D. It is natural to investigate whether there is a similar relationship in digraphs. One such attempt was made in Theorem 1.4 of [3]. It is unfortunately that, due to the misunderstanding of a result of Frank (mistakenly quoted as a result of Edmonds in Theorem 3.1 of [3]), the proof of Theorem 1.4 of [3] is false. Therefore, the problem of investigating the relationship between λ(d) and τ(d) remains unanswered. Problem 4. Is there a similar relationship between λ(d) and τ(d) for a digraph D as the Theorem 1.6 for graphs G? For a graph G, we follow [2] to use c(g) to denote the number of components of G. Then the strength η(g) and fractional arboricity γ(g) of G are defined as { } { } X X η(g) = min, and γ(g) = max, X E(G) c(g X) c(g) X E(G) V (G[X]) c(g[x]) where the minimum and maximum are taken over all subsets X such that the corresponding denominators are not zero. In [4], it has been indicated that the well-known spanning tree packing theorem of Nash-Williams [27] and Tutte [32] can be restated as the following. Theorem 1.7. A nontrivial graph G has k edge-disjoint spanning trees if and only if η(g) k. In [4], an attempt to obtain a fractional version of Theorem 1.7 was made, and the following characterization is proved. Theorem 1.8. ( Catlin et al. Theorem 6 of [4]) Let G be a connected graph. The following are equivalent. (i) γ(g)( V (G) 1) = E(G). (ii) η(g)( V (G) 1) = E(G). (iii) η(g) = γ(g). (iv) For any integers s t > 0 with γ(g) = s t, G has s spanning trees T 1, T 2,, T s such that every edge e E(G) is in exactly t members in the multiset {T 1, T 2,, T s }. (v) For any integers s t > 0 with η(g) = s t, G has s spanning trees T 1, T 2,, T s such that every edge e E(G) is in exactly t members in the multiset {T 1, T 2,, T s }. A graph G satisfying Theorem 1.8(i) is called a uniformly dense graph. These results have been applied in [8] to give short proofs for some results on higher-order edge toughness of a graph in [6]; and to obtained characterization of of minimal graphs whose edge-connectivity equals the 4

5 spanning tree packing number in [11]. The characterizations of uniformly dense graphs have also been applied to many other studies, such as cyclic base orderings [14] and the problem of packing hypertrees [10]. It is natural to consider the study on uniformly dense digraphs analogous to those characterizations in [4]. Problem 5. What are the characterizations of uniformly dense digraphs? Given a graph G, Matula [24 26] first studied the quantity κ (G) = max{κ (H) : H G}. He called κ (G) the strength of G. Mader [23] considered an extremal problem related to κ (G). For an integer k > 0, a simple graph G with V (G) k + 1 is k-maximal if κ (G) k but for any edge e E(G c ), κ (G + e) > k. In [23], Mader proved the following. Theorem 1.9. (Mader [23]) If G is a k-maximal graph on n > k 1 vertices, then ( ) k E(G) (n k)k +. 2 Furthermore, this bound is best possible. It has been noted that being a k-maximal graph requires a certain level of edge density. Towards this direction, the following was proved in Theorem (Lai, Theorem 2 of [18]) If G is a k-maximal graph on n > k vertices, then ( ) k n E(G) (n 1)k 2 k + 2. Furthermore, this bound is best possible. It is natural to consider extending the theorems above to digraphs. Problem 6. What is the corresponding result in digraphs? 2 Progress Towards those motivations on both graph coloring and digraph connectivity problems, some results have been obtained. 2.1 Coloring Problems For Problem 1, we consider a graph G without K 4 -minor with = (G). Let r 1 be an integer, define f(, r) = max{r, /2 } + /

6 Theorem 2.1. Let G be a K 4 -minor free graph with = (G), and r 1 be an integer. Then χ l L,r (G) f(, r). For Problem 2, let H 1, H 2 be subgraphs of G. Then the unordered pair (H 1, H 2 ) is a separation of G if E(H 1 ) E(H 2 ) = E(G) and G has no edge shared in H 1 and H 2. Let G be a graph with a separation (H 1, H 2 ). Let K be a complete graph with V (K) = V (H 1 ) V (H 2 ) and with no edges in common with H 1 and H 2. Let G i = H i K (i = 1, 2). Then G is the clique-sum of G 1 and G 2. Proposition 2.2. Let G = G 1 t G 2, where G has a separation (H 1, H 2 ), K t is a complete graph of order t with V (K t ) = V (H 1 ) V (H 2 ), and G i = H i K t (i = 1, 2). If E(K) E(G), then (i)χ 2 (G) max{χ 2 (G 1 ), χ 2 (G 2 )}; (ii)if G 2 = C 5 and G = G 1 2 G 2, then χ 2 (G) max{χ 2 (G 1 ), 3}. In particular, χ 2 (C 5 2 C 5) = 4. Theorem 2.3. Let G = G 1 2 G 2 with V (G 1 ) V (G 2 ) = {x, y}. If d Gi (x), d Gi (y) 2(i = 1, 2), then χ 2 (G) max{χ 2 (G 1 ), χ 2 (G 2 ), 5}. Theorem 2.4. (Hall [13]). Let G be a graph without K 3,3 minor. One of the followings must hold: (i) G is a planar graph; (ii) G = K 5, or (iii) G can be constructed recursively by the i-sum of planar graphs and copies of K 5 where i {1, 2}. Then applying Theorem 2.3 and Hall s Theorem, we find a best possible upper bound of χ 2 (G) for all connected graphs G without a K 3,3 -minor. Corollary 2.5. Let G be a connected graph without K 3,3 minor. Then χ 2 (G) 5. For Problem 3, we used the charge and discharge method to obtain the following. Theorem 2.6. Let G be a graph with maximum degree. max{ /2 + 1, r + 1}, r 6; (i) If Mad(G) < 12 5, then χl L,r (G) max{ /2 + 1, 7}, r = 5; max{ /2 + 1, 6}, 1 r 5. (ii) If Mad(G) < 5 2, then χl L,r (G) max{ /2 + 1, r + 2, 7}. (iii) If Mad(G) < 8 3, then χl L,r (G) max{ /2 + 2, r + 3, 8}. 2.2 Digraph Connectivity Problems For Problem 4, we introduced some definitions and notations first. 6

7 Definition 2.1. Let T be an oriented tree with a fixed vertex r V (T ). T is an out-arborescence rooted at r (or an r + -arborescence) if d T (r) = 0 and for any v V (T ) r, d T (v) = 1; T is an in-arborescence rooted at r (or an r -arborescence) if d + T (r) = 0 and for any v V (T ) r, (v) = 1. In either case, r is called the root of T. d + T An out-arborescence is also called an arborescence, and an r + -arborescence is also called an r-arborescence. For any function f : V (D) R + and any subset S V (D), define f(s) = v S f(v). Definition 2.2. Let k s 1 be integers. (i) Let A 1 (k, s) be the family of digraphs such that D A 1 (k, s) if and only if for any S V (D) with S s, D has k arc-disjoint spanning out-arborescences whose roots are in S. (ii) Let A 2 (k, s) be the family of digraphs such that D A 2 (k, s) if and only if for any S V (D) with S s, D has k arc-disjoint spanning in-arborescences whose roots are in S. Note that A(k) = A 1 (k, 1). For each i {1, 2}, define f i (n, k, s) = min{ A(D) : D A i (k, s) is simple with V (D) = n}. (1) We proved the following theorem. Theorem 2.7. For each i {1, 2}, and for any integers k, n, s with n k + 1, and n > s > 0, f i (n, k, s) = nk. (2) For Problem 5, for a digraph D with n = V (D) 2, let P = P(D) = {(X 1, X 2,, X m ) where X 1, X 2, X p are disjoint nonempty subsets of V (D)}. We define and γ(d) = η(d) = min (X 1,X 2, X p) P(D),p>1 max (X 1,X 2, X p) P(D),p<n { p i=1 δ D (X i) p 1 { A(D) p i=1 δ D (X i) n p }, (3) }. (4) Frank proved two theorems analogues to the spanning tree packing theorem of Nash-Williams [27] and Tutte [32] and the forest covering theorem of Nash-William [28]. A branching of a digraph D is a sub digraph B of D such that every weakly connected component of B is an arborescence. Theorem 2.8. (Frank [9]) Let D be a nontrivial digraph. The following are equivalent. (i) D is has k-arc-disjoint spanning arborescences. (ii) η(d) k. Theorem 2.9. (Frank [9]) Let D be a nontrivial digraph. The following are equivalent. (i) D has k branchings such that every arc of D is in at least one of them. (ii) γ(d) k. 7

8 We proved the following. Lemma Let D be a digraph, and k, t > 0 be integers. (i) D has k spanning arborescences such that every e A(D) lies in at most t of them if and only if η(d) k t. (ii) D has k branchings such that every e A(D) lies in at least t of them if and only if γ(d) k t. Theorem Let D be a weakly connected nontrivial digraph on n vertices. The followings are equivalent. (A digraph satisfying any one of the following is called a uniformly dense digraph). (i) A(D) = η(d)(n 1). (ii) A(D) = γ(d)(n 1). (iii) η(d) = γ(d). (iv) For any integers s t > 0 such that γ(d) = s t. there exists an integer t > 0 such that D has a family F = {T 1, T 2,, T s } of spanning arborescences such that every e A(D) lies in exactly t members in F. (v) For any integers s t > 0 such that η(d) = s t. there exists an integer t > 0 such that D has a family F = {T 1, T 2,, T s } of spanning arborescences such that every e A(D) lies in exactly t members in F. For Problem 6, we define λ(d) = max{λ(h) : H D}. Let k 0 be an integer. A simple digraph D with V (D) k + 1 is k-maximal if λ(d) k but for any arc e A(D c ), λ(d + e) k + 1. For positive integer n and k with n k + 1, define D(n, k) = {D : D is a simple digraph with V (D) = n and D is k-maximal}. We proved the following theorem. Theorem Let n, k be integers with n > k Then for any D D(n, k), we have ( ) ( ( )) n n k + 2 A(D) + (n 1)k + 2 k k. (5) 2 Furthermore, the bound is best possible. 3 Future Work 3.1 Coloring Problems We will continue working towards an answer to Conjecture 1.2, as well as list r-hued colorings and linear list r-hued colorings for planar graphs and, more generally, graphs embedded in surfaces with higher genus. There are two fundamental conjectures in graph coloring problems. 8

9 Conjecture 3.1. (Hadwiger [12]) For every integer t 0, every graph with no K t+1 minor is t-colorable. Conjecture 3.2. (Chartrand, Geller, Hedetniemi [5]; Woodall [34]) For integers r s 1, every graph without a K r,s -minor is (r + s 1)-colorable. We will also work on the r-hued coloring versions of these conjectures. 3.2 Digraph Arc-strong Connectivity Problems In [21], Liu et al. considered a network reinforcement problem: for a given integer k and a given graph G, determine the smallest number of additional edges X such that G + X has k-edgedisjoint spanning trees. By using a decomposition theorem, this problem was solved by Liu et al. in [21]. The matroid version of this problem was answered in [20]. As arc sets inducing acyclic subdigraphs may not be independent set of a matroid on the arc set of a digraph, the corresponding reinforcement problem cannot be implied by the main result in [20]. Thus we are to work on this problem: Given an integer k and a digraph D, determine the smallest number of additional arc sets X such that D + X has the property that for any v V (D), D + X has k arc-disjoint spanning r + -arborescences. Payan in [29] considered another type of network reinforcement problem: Given an integer k, design a nonnegative integral function f on all graphs which measures how close a graph G is to having k-edge-disjoint spanning trees with f(g) = 0 being the state that G has k-edge-disjoint spanning trees. Payan conjectured in [29] that if E(G) = k( V (G) 1), then there exists an edge e E(G) and an edge e E(G c ) (G c is the complement of G) such that f(g e + e ) < f(g). Thus any such graph can be reinforced to having k-edge-disjoint spanning trees using a finite number of edge swappings. This conjecture was proved in [19]. We want to work on the corresponding digraph problem: For any integer k > 0, design a nonnegative integral function f on (strict) digraphs which measures how close a digraph D is to having the property that for any v V (D), D has k- arc-disjoint spanning r + -arborescences with f(d) = 0 being the state that D itself has such a property. We want to prove that for any (strict) digraph D, there exists an arc e A(D) and an arc e A(D c ) (D c is the complement of D) such that f(d e + e ) < f(d). References [1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd Edition, Springer-Verlag, London, [2] J. A. Bondy and U. S. R. Murty, Graph Theory. Springer, New York,

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