Linear Choosability of Sparse Graphs

We study the linear list chromatic number, denoted \lcl(G), of sparse graphs. The maximum average degree of a graph $G$, denoted \mad(G), is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $\Delta(G)$ satisfies \lcl(G)\ge \ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if \mad(G)<12/5 and $\Delta(G)\ge 3$, then \lcl(G)=\ceil{\Delta(G)/2}+1, and we give an infinite family of examples to show that this result is best possible; (2) if \mad(G)<3 and $\Delta(G)\ge 9$, then \lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to show that the bound on \mad(G) cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring

Linear and 2-Frugal Choosability of Graphs of Small Maximum Average Degree

International audienceA proper vertex colouring of a graph G is 2-frugal (resp. linear) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph G is 2-frugally (resp. linearly) L-colourable if for a given list assignment L : V(G) → N, there exists a 2-frugal (resp. linear) colouring c of G such that c(v) ∈ L(v) for all v ∈ V (G). If G is 2-frugally (resp. linearly) L-list colourable for any list assignment such that |L(v)| ≥ k for all v ∈ V (G), then G is 2-frugally (resp. linearly) k-choosable. In this paper, we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree

A Study on Graph Coloring and Digraph Connectivity

This dissertation focuses on coloring problems in graphs and connectivity problems in digraphs. We obtain the following advances in both directions.;1. Results in graph coloring. For integers k,r \u3e 0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d( v) is adjacent to vertices with at least min{lcub}d( v),r{rcub} different colors. The r-hued chromatic number, denoted by chir(G ), is the smallest integer k for which a graph G has a (k,r)-coloring.;For a k-list assignment L to vertices of a graph G, a linear (L,r)-coloring of a graph G is a coloring c of the vertices of G such that for every vertex v of degree d(v), c(v)∈ L(v), the number of colors used by the neighbors of v is at least min{lcub}dG(v), r{rcub}, and such that for any two distinct colors i and j, every component of G[c --1({lcub}i,j{rcub})] must be a path. The linear list r-hued chromatic number of a graph G, denoted chiℓ L,r(G), is the smallest integer k such that for every k-list L, G has a linear (L,r)-coloring. Let Mad( G) denotes the maximum subgraph average degree of a graph G. We prove the following. (i) If G is a K3,3-minor free graph, then chi2(G) ≤ 5 and chi3(G) ≤ 10. Moreover, the bound of chi2( G) ≤ 5 is best possible. (ii) If G is a P4-free graph, then chir(G) ≤q chi( G) + 2(r -- 1), and this bound is best possible. (iii) If G is a P5-free bipartite graph, then chir( G) ≤ rchi(G), and this bound is best possible. (iv) If G is a P5-free graph, then chi2(G) ≤ 2chi(G), and this bound is best possible. (v) If G is a graph with maximum degree Delta, then each of the following holds. (i) If Delta ≥ 9 and Mad(G) \u3c 7/3, then chiℓL,r( G) ≤ max{lcub}lceil Delta/2 rceil + 1, r + 1{rcub}. (ii) If Delta ≥ 7 and Mad(G)\u3c 12/5, then chiℓ L,r(G)≤ max{lcub}lceil Delta/2 rceil + 2, r + 2{rcub}. (iii) If Delta ≥ 7 and Mad(G) \u3c 5/2, then chi ℓL,r(G)≤ max{lcub}lcei Delta/2 rceil + 3, r + 3{rcub}. (vi) If G is a K 4-minor free graph, then chiℓL,r( G) ≤ max{lcub}r,lceilDelta/2\rceil{rcub} + lceilDelta/2rceil + 2. (vii) Every planar graph G with maximum degree Delta has chiℓL,r(G) ≤ Delta + 7.;2. Results in digraph connectivity. For a graph G, let kappa( G), kappa\u27(G), delta(G) and tau( G) denote the connectivity, the edge-connectivity, the minimum degree and the number of edge-disjoint spanning trees of G, respectively. Let f(G) denote kappa(G), kappa\u27( G), or Delta(G), and define f¯( G) = max{lcub}f(H): H is a subgraph of G{rcub}. An edge cut X of a graph G is restricted if X does not contain all edges incident with a vertex in G. The restricted edge-connectivity of G, denoted by lambda2(G), is the minimum size of a restricted edge-cut of G. We define lambda 2(G) = max{lcub}lambda2(H): H ⊂ G{rcub}.;For a digraph D, let kappa;(D), lambda( D), delta--(D), and delta +(D) denote the strong connectivity, arc-strong connectivity, minimum in-degree, and out-degree of D, respectively. For each f ∈ {lcub}kappa,lambda, delta--, +{rcub}, define f¯(D) = max{lcub} f(H): H is a subdigraph of D{rcub}.;Catlin et al. in [Discrete Math., 309 (2009), 1033-1040] proved a characterization of kappa\u27(G) in terms of tau(G). We proved a digraph version of this characterization by showing that a digraph D is k-arc-strong if and only if for any vertex v in D, D has k-arc-disjoint spanning arborescences rooted at v. We also prove a characterization of uniformly dense digraphs analogous to the characterization of uniformly dense undirected graphs in [Discrete Applied Math., 40 (1992) 285--302]. (Abstract shortened by ProQuest.)