A lower bound for partial list colorings

Let G be an n-vertex graph with list-chromatic number χ_ℓ. Suppose that each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas [1] conjecture that at least t_n/χ_ℓ vertices can be colored from these lists. We prove a lower bound for the number of colorable vertices. As a corollary, we show that at least of the conjectured number can be colored. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 390–393, 1999     

一类完全多部图的色可选择性

如果一个图G的选择数等于它的色数,则称该图G是色可选择的．在2002年, Ohba给出如下猜想:每一个顶点个数小于等于2X(G) 1的图G是色可选择的．容易发现Ohba猜想成立的条件是当且仅当它对完全多部图成立,但是目前只是就某些特殊的完全多部图的图类证明了Ohba猜想的正确性．在本文我们证明图K6,3,2*(k-6),1*4(k≥6)是色可选择的,从而对图K6,3,2*(k-6),1*4(k≥6)和它们的所有完全k-部子图证明了Ohba猜想成立．     

On a list coloring conjecture of Reed

We construct graphs with lists of available colors for each vertex, such that the size of every list exceeds the maximum vertex‐color degree, but there exists no proper coloring from the lists. This disproves a conjecture of Reed. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 106–109, 2002     

Acyclic list 7‐coloring of planar graphs

The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002     

On (4, 2)‐Choosable Graphs

A graph G is called ‐choosable if for any list assignment L that assigns to each vertex v a set of a permissible colors, there is a b‐tuple L‐coloring of G . An (a , 1)‐choosable graph is also called a‐choosable. In the pioneering article on list coloring of graphs by Erd?s et al.  2 , 2‐choosable graphs are characterized. Confirming a special case of a conjecture in  2 , Tuza and Voigt  3 proved that 2‐choosable graphs are ‐choosable for any positive integer m . On the other hand, Voigt 6 proved that if m is an odd integer, then these are the only ‐choosable graphs; however, when m is even, there are ‐choosable graphs that are not 2‐choosable. A graph is called 3‐choosable‐critical if it is not 2‐choosable, but all its proper subgraphs are 2‐choosable. Voigt conjectured that for every positive integer m , all bipartite 3‐choosable‐critical graphs are ‐choosable. In this article, we determine which 3‐choosable‐critical graphs are (4, 2)‐choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer k such that for any positive integer m , every bipartite 3‐choosable‐critical graph is ‐choosable. Moving beyond 3‐choosable‐critical graphs, we present an infinite family of non‐3‐choosable‐critical graphs that have been shown by computer analysis to be (4, 2)‐choosable. This shows that the family of all (4, 2)‐choosable graphs has rich structure.     

关于Reed列表染色猜想的一些结果

构造了一个图G,给G的每个顶点v一个颜色列表,使得每个列表Lv的大小至少为每个顶点v的邻域NG(v)与每个Vc交集的最大数目,但是这个图不存在一个正常的列表染色,从而推翻了R eed的一个猜想.     

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

Equitable List Coloring of Graphs with Bounded Degree

A graph G is equitably k‐choosable if for every k‐list assignment L there exists an L‐coloring of G such that every color class has at most vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably ‐choosable. In particular, we confirm the conjecture for and show that every graph with maximum degree at most r and at least r³ vertices is equitably ‐choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings.     

Sharp Dirac's theorem for DP‐critical graphs

Correspondence coloring, or DP‐coloring, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that satisfy Dirac's bound with equality.     

A Relaxed Version of the Erdős–Lovász Tihany Conjecture

The Erd?s–Lovász Tihany conjecture asserts that every graph G with ) contains two vertex disjoint subgraphs G ₁ and G ₂ such that and . Under the same assumption on G , we show that there are two vertex disjoint subgraphs G ₁ and G ₂ of G such that (a) and or (b) and . Here, is the chromatic number of is the clique number of G , and col(G ) is the coloring number of G .     

Circular choosability of graphs

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) χ_{c, l}(G) of a graph G and prove that they are equivalent. Then we prove that for any graph G, χ_{c, l}(G) ≥ χ_l(G) ? 1. Examples are given to show that this bound is sharp in the sense that for any ? 0, there is a graph G with χ_{c, l}(G) > χ_l(G) ? 1 + ?. It is also proved that k‐degenerate graphs G have χ_{c, l}(G) ≤ 2k. This bound is also sharp: for each ? < 0, there is a k‐degenerate graph G with χ_{c, l}(G) ≥ 2k ? ?. This shows that χ_{c, l}(G) could be arbitrarily larger than χ_l(G). Finally we prove that if G has maximum degree k, then χ_{c, l}(G) ≤ k + 1. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 210–218, 2005     

Total choosability of multicircuits II

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this paper, the method of Alon and Tarsi is used to prove that all multicircuits of even order, and some regular and near‐regular multicircuits of odd order have total choosability (i.e., list total chromatic number) equal to their ordinary total chromatic number. This completes the proof that every multicircuit has total choosability equal to its total chromatic number. In the process, the total chromatic numbers of all multicircuits are determined. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 44–67, 2002     

Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .     

Total weight choosability of graphs

A graph G = (V, E) is called (k, k′)‐total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f: V∪E→? such that f(y)∈L(y) for any y∈V∪Eand for any two adjacent vertices x, x′, . We conjecture that every graph is (2, 2)‐total weight choosable and every graph without isolated edges is (1, 3)‐total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K₂ are (1, 3)‐total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)‐total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)‐total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)‐total weight choosable as well as (1, 3)‐total weight choosable. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:198‐212, 2011     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

For a positive integer k, a k‐coloring of a graph is a mapping such that whenever . The Coloring problem is to decide, for a given G and k, whether a k‐coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k‐Coloring instead. We survey known results on the computational complexity of Coloring and k‐Coloring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.     

Total choosability of multicircuits I

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch′′(C) equal to its ordinary total chromatic number χ′′(C). In the present paper, the kernel method is used to prove this for every multicircuit that has at least two vertices with degree less than its maximum degree Δ. The result is also proved for every multicircuit C for which χ′′(C)≥Δ+2. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 26–43, 2002     

List Colorings with Distinct List Sizes,the Case of Complete Bipartite Graphs

Let be a function on the vertex set of the graph . The graph G is f‐choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, , is the minimum of , over all functions f such that G is f‐choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1–33, Random Struct Algor 16 (2000), 364–368) that if G has average degree d, then the usual choice number is at least , so they grow simultaneously. In this article, we show that can be bounded while the minimum degree . Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph .     

List circular coloring of trees and cycles

Suppose G=(V, E) is a graph and p ≥ 2q are positive integers. A (p, q)‐coloring of G is a mapping ?: V → {0, 1, …, p‐1} such that for any edge xy of G, q ≤ |?(x)‐?(y)| ≤ p‐q. A color‐list is a mapping L: V → ({0, 1, …, p‐1}) which assigns to each vertex v a set L(v) of permissible colors. An L‐(p, q)‐coloring of G is a (p, q)‐coloring ? of G such that for each vertex v, ?(v) ∈ L(v). We say G is L‐(p, q)‐colorable if there exists an L‐(p, q)‐coloring of G. A color‐size‐list is a mapping ? which assigns to each vertex v a non‐negative integer ?(v). We say G is ?‐(p, q)‐colorable if for every color‐list L with |L(v)| = ?(v), G is L‐(p, q)‐colorable. In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be ?‐(p, q)‐colorable. For each cycle C and for each positive integer k, we present a condition on ? which is sufficient for C to be ?‐(2k+1, k)‐colorable, and the condition is sharp. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 249–265, 2007