Sum List Coloring Block Graphs

A graph 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. We characterize f-choosable functions for block graphs (graphs in which each block is a clique, including trees and line graphs of trees). The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f. The sum choice number of any graph is at most the number of vertices plus the number of edges. We show that this bound is tight for block graphs.Acknowledgments. Partially supported by a grant from the Reidler Foundation. The author would like to thank an anonymous referee for useful comments.     

Sum choice numbers of some graphs

Let f be a function assigning list sizes to the vertices of a graph G. The sum choice number of G is the minimum ∑_v∈V(G)f(v) such that for every assignment of lists to the vertices of G, with list sizes given by f, there exists proper coloring of G from the lists. We answer a few questions raised in a paper of Berliner, Bostelmann, Brualdi, and Deaett. Namely, we determine the sum choice number of the Petersen graph, the cartesian product of paths , and the complete bipartite graph K_3,n.     

Partial Online List Coloring of Graphs

For a graph G, let be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s‐choosable and , then . In this article, we consider the online version of this conjecture. Let be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s‐choosable and then . This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers , . As a consequence, if s is a multiple of t, then . We also prove that if G is online s‐choosable and , then and for any , .     

外平面图的全染色与列表全染色

本文证明了,如果G是满足条件Δ(G)≥4的外平面图,则x_T~L(G)=Δ(G) 1,同时对Δ(G)=3给出了XT(G)=Δ(G) 1的简短的新证明,从而蕴含Δ(G)≥3时,XT(G)=Δ(G) 1,其中XT(G)是G的点边全色数,x_T~L(G)是G的点边列表全色数。     

Acyclic List Edge Coloring of Graphs

A proper edge coloring of a graph is said to be acyclic if any cycle is colored with at least three colors. An edge-list L of a graph G is a mapping that assigns a finite set of positive integers to each edge of G. An acyclic edge coloring ? of G such that for any is called an acyclic L-edge coloring of G. A graph G is said to be acyclically k-edge choosable if it has an acyclic L‐edge coloring for any edge‐list L that satisfies for each edge e. The acyclic list chromatic index is the least integer k such that G is acyclically k‐edge choosable. We develop techniques to obtain bounds for the acyclic list chromatic indices of outerplanar graphs, subcubic graphs, and subdivisions of Halin graphs.     

The Entire Coloring of Series-Parallel Graphs

The entire chromatic number X_(vef)(G) of a plane graph G is the minimal number of colors needed for coloring vertices, edges and faces of G such that no two adjacent or incident elements are of the same color. Let G be a series-parallel plane graph, that is, a plane graph which contains no subgraphs homeomorphic to K_(4-) It is proved in this paper that X_(vef)(G)≤max{8, △(G) 2} and X_(vef)(G)=△ 1 if G is 2-connected and △(G)≥6.     

系列平行图的边面染色

一个平面图G的边面色数xef(G)是指对G的边和面进行染色所用最少的颜色数目,并同时使得相邻或相关联的两个元素间染不同颜色.若G是一个系列平行图,也就是不含K_4的剖分作为子图的平面图,则有Xef(G)≤max{7,△(G) 1};同时如果G还是2-连通的且△(G)>6,则有Xef(G)=△.     

Sum choice number of generalized θ-graphs

Let $G=\left(VE\right)$ be a simple graph and for every vertex $v\in V$ let $L\left(v\right)$ be a set (list) of available colors. $G$ is called $L$-colorable if there is a proper coloring $\phi$ of the vertices with $\phi \left(v\right)\in L\left(v\right)$ for all $v\in V$. A function $f:V\to \mathbb{N}$ is called a choice function of $G$ and $G$ is said to be $f$-list colorable if $G$ is $L$-colorable for every list assignment $L$ choice function is defined by $size\left(f\right)={\sum }_{v\in V}f\left(v\right)$ and the sum choice number ${\chi }_{sc}\left(G\right)$ denotes the minimum size of a choice function of $G$.Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then.For $r\ge 3$ a generalized ${\theta }_{k_1{k}_2\dots k_r}$-graph is a simple graph consisting of two vertices $v_1$ and $v_2$ connected by $r$ internally vertex disjoint paths of lengths $k_1,k_2,\dots ,k_r$ $(k_1\le k_2\le ?\le k_r)$.In 2014, Carraher et al. determined the sum-paintability of all generalized $\theta$-graphs which is an online-version of the sum choice number and consequently an upper bound for it.In this paper we obtain sharp upper bounds for the sum choice number of all generalized $\theta$-graphs with $k_1\ge 2$ and characterize all generalized $\theta$-graphs $G$ which attain the trivial upper bound $\left|V\left(G\right)\right|+\left|E\left(G\right)\right|$.     

若干联图的均匀全染色

图$G(V,E)$的全色数 $\chi_{t}(G)$就是将$V\bigcup E$分成彼此不相交的全独立分割集的最小个数。 如果任何两个$V\bigcup E$的全独立分割集的元素数目相差不超过1，那么 $V \bigcup E$的全独立分割集的最小个数就称为图$G$的均匀全色数，记为$\chi_{et}(G)$。 在本文中我们给出了当 $m \geq n \geq 3$ 时 $W_m\bigvee K_n$，$F_m \bigvee K_n$及$S_m \bigvee K_n$ 的均匀全色数.     

List coloring of Cartesian products of graphs

A well-established generalization of graph coloring is the concept of list coloring. In this setting, each vertex v of a graph G is assigned a list L(v) of k colors and the goal is to find a proper coloring c of G with c(v)∈L(v). The smallest integer k for which such a coloring c exists for every choice of lists is called the list chromatic number of G and denoted by χ_l(G).We study list colorings of Cartesian products of graphs. We show that unlike in the case of ordinary colorings, the list chromatic number of the product of two graphs G and H is not bounded by the maximum of χ_l(G) and χ_l(H). On the other hand, we prove that χ_l(G×H)?min{χ_l(G)+col(H),col(G)+χ_l(H)}-1 and construct examples of graphs G and H for which our bound is tight.     

List Point Arboricity of Dense Graphs

Let G be a simple graph. The point arboricity ρ(G) of G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph. The list point arboricity ρ _l (G) is the minimum k so that there is an acyclic L-coloring for any list assignment L of G which |L(v)| ≥ k. So ρ(G) ≤ ρ _l (G) for any graph G. Xue and Wu proved that the list point arboricity of bipartite graphs can be arbitrarily large. As an analogue to the well-known theorem of Ohba for list chromatic number, we obtain ρ _l (G + K _n ) = ρ(G + K _n ) for any fixed graph G when n is sufficiently large. As a consequence, if ρ(G) is close enough to half of the number of vertices in G, then ρ _l (G) = ρ(G). Particularly, we determine that , where K _2(n) is the complete n-partite graph with each partite set containing exactly two vertices. We also conjecture that for a graph G with n vertices, if then ρ _l (G) = ρ(G). Research supported by NSFC (No.10601044) and XJEDU2006S05.     

关于一类二部图的均匀邻点可区别全染色

若图的邻点可区别全染色的各色所染元素数之差不超过1,则称该染色法为图的均匀邻点可区别全染色,而所用的最少颜色数称为该图的均匀邻点可区别全色数.本文给出了一类二部图的均匀邻点可区别全染色数.     

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex.A proper total-k-coloring of G is called neighbor sum distinguishing if∑_c(u)≠∑_c(v)for each edge uv∈E(G),where∑_c(v)denote the sum of the color of a vertex v and the colors of edges incident with v.The least number k needed for such a total coloring of G,denoted byχ∑"is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjecturedχ∑"(G)≤Δ(G)+3 for any simple graph with maximum degreeΔ(G).By using the famous Combinatorial Nullstellensatz,we prove that above conjecture holds for any triangle free IC-planar graph with△(G)≥7.Moreover,it holds for any triangle free planar graph withΔ(G)≥6.     

List Coloring with a Bounded Palette

Král' and Sgall (J Graph Theory 49(3) (2005), 177–186) introduced a refinement of list coloring where every color list must be subset to one predetermined palette of colors. We call this ‐choosability when the palette is of size at most ? and the lists must be of size at least k . They showed that, for any integer , there is an integer , satisfying as , such that, if a graph is ‐choosable, then it is C‐choosable, and asked if C is required to be exponential in k . We demonstrate it must satisfy . For an integer , if is the least integer such that a graph is ‐choosable if it is ‐choosable, then we more generally supply a lower bound on , one that is super‐polynomial in k if , by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on that improve on earlier bounds if .     

若干图的点强全着色

图G(V,E)的一正常k-全着色σ称为G(V,E)的一个k-点强全着色,当且仅当v∈V(G),N[v]中的元素着不同颜色,其中N[v]={u|vu∈E(G)}∪{v}.并且vχsT(G)=min{k|存在G的一个k-点强全着色}称为G(V,E)的点强全色数.本文得到了一些特殊图的点强全色数χvTs(G),并提出猜想:对于简单图G,有k(G)≤χvTs(G)≤k(G)+1,这里k(G)表示图G中所有顶点间距离不超过2的点集的最大顶点数.     

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.     

外平面图的弱边面染色

假设G是一个平面图.如果e1和e2是G中两条相邻边且在关联的面的边界上连续出现,那么称e1和e2面相邻.图G的一个弱边面κ-染色是指存在映射π:E∪F→{1,…,κ},使得任意两个相邻面、两条面相邻的边以及两个相关联的边和面都染不同的颜色.若图G有一个弱边面κ-染色,则称G是弱边面κ-可染的.平面图G的弱边面色数是指G是弱边面κ-可染的正整数κ的最小值,记为χef(G).2016年,Fabrici等人猜想:每个无环且无割边的连通平面图是弱边面5-可染的.本文证明了外平面图满足此猜想,即:外平面图是弱边面5-可染的.     

A GRASP for Coloring Sparse Graphs

We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs with edge probability 0.1 having from 50 to 500 vertices. Empirical results indicate that the proposed GRASP implementation compares favorably to classical heuristics and implementations of simulated annealing and tabu search. GRASP is also found to be competitive with a genetic algorithm that is considered one of the best currently available for graph coloring.     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

最大度为4的图的无圈列表边染色

对于图G=(V(G),E(G)),如果一个映射φ:E(G)→{1,2,…,k},使得G中任意相邻的两边e₁,e₂满足φ(e₁)≠φ(e₂),并且G中不含有双色圈,则称φ为G的一个无圈边染色.对于给定的列表分配L={L(e)|e∈E(G)},如果存在图G的一个无圈边染色φ,使得对于任意边e∈E(G),均有φ(e)∈L(e),则称染色φ为G的一个无圈L-边染色.如果对于任意的列表分配L,当对所有的边e∈E(G)满足|L(e)|≥k时,图G均存在无圈L-边染色,那么称G是无圈k-边可选的.使图G无圈k-边可选的最小的正整数k,称为G的无圈列表边色数,用a’_l(G)表示.本文证明了对于最大度△≤4的连通图G,如果|E(G)|≤2|V(G)|-1,则a’_l(G)≤6,扩展了Basavaraju和Chandran文[J.Graph Theory,2009,61(3):192-209]的结果.