Optimal acyclic edge‐coloring of cubic graphs

An acyclic edge‐coloring of a graph is a proper edge‐coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge‐coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K₄ or K_3,3; the acyclic chromatic index of K₄ and K_3,3 is 5. This result has previously been published by Fiam?ík, but his published proof was erroneous.     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let Forb*(H) be the class of graphs with no induced subdivision of H. It was conjectured in [J Graph Theory, 24 (1997), 297–311] that, for every graph H, there is a function f_H: ?→? such that for every graph G∈Forb*(H), χ(G)≤f_H(ω(G)). We prove this conjecture for several graphs H, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex‐disjoint pendant edges), and what we call a “necklace,” that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:49–68, 2012     

Polynomial algorithm for sharp upper bound of rainbow connection number of maximal outerplanar graphs

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2011) [5] proved that computing rc(G) is NP-hard and deciding if rc(G)=2 is NP-complete. When edges of G are colored with fixed number k of colors, Kratochvil [6] proposed a question: what is the complexity of deciding whether G is rainbow connected? is this an FPT problem? In this paper, we prove that any maximal outerplanar graph is k rainbow connected for suitably large k and can be given a rainbow coloring in polynomial time.     

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.     

平面图点荫度的一个局部条件

图G的点荫度va(G)是指V(G)的最小划分数,使得每个点划分集的导出子图是一个森林.众所周知,平面图G的点荫度至多为3.2008年,Raspaud和王维凡证明了:若G是不含κ-圈(κ∈{3,4,5,6})的平面图,则va(G)≤2.本文推广了上述结果,得到了点荫度至多为2的一个局部条件,即若平面图G的每一个点都不同时与3-圈、4-圈、5-圈和6-圈关联,则va(G)≤2.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

List 3-dynamic coloring of graphs with small maximum average degree

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list$r$-dynamic chromatic number of a graph $G$ is denoted by $c{h}_r^d\left(G\right)$.Recently, Loeb et al. (0000) showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. (0000) studied the maximum average condition to have ${\chi }_3^d\left(G\right)\le 4,5$, or $6$. On the other hand, Song et al. (2016) showed that if $G$ is planar with girth at least 6, then ${\chi }_r^d\left(G\right)\le r+5$ for any $r\ge 3$.In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $c{h}_3^d\left(G\right)\le 6$ if $mad\left(G\right)<\frac{18}{7}$, $c{h}_3^d\left(G\right)\le 7$ if $mad\left(G\right)<\frac{14}{5}$, and $c{h}_3^d\left(G\right)\le 8$ if $mad\left(G\right)<3$. All of the bounds are tight.     

Hall number for list colorings of graphs: Extremal results

Given a graph G=(V,E) and sets L(v) of allowed colors for each v∈V, a list coloring of G is an assignment of colors φ(v) to the vertices, such that φ(v)∈L(v) for all v∈V and φ(u)≠φ(v) for all uv∈E. The choice number of G is the smallest natural number k admitting a list coloring for G whenever |L(v)|≥k holds for every vertex v. This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L(v). (On complete graphs, this condition is equivalent to the well-known one in Hall’s Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n>3 may make the Hall number decrease by as much as n−3. This estimate is tight for all n. Tightness is deduced from the upper bound that every graph of order n has Hall number at most n−2. We also characterize the cases of equality; for n≥6 these are precisely the graphs whose complements are K₂∪(n−2)K₁, P₄∪(n−4)K₁, and C₅∪(n−5)K₁. Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161-182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227-245] as a function of maximum degree.     

Color-bounded hypergraphs, IV: Stable colorings of hypertrees

We consider vertex colorings of hypergraphs in which lower and upper bounds are prescribed for the largest cardinality of a monochromatic subset and/or of a polychromatic subset in each edge. One of the results states that for any integers s≥2 and a≥2 there exists an integer f(s,a) with the following property. If an interval hypergraph admits some coloring such that in each edge E_i at least a prescribed number s_i≤s of colors occur and also each E_i contains a monochromatic subset with a prescribed number a_i≤a of vertices, then a coloring with these properties exists with at most f(s,a) colors. Further results deal with estimates on the minimum and maximum possible numbers of colors and the time complexity of determining those numbers or testing colorability, for various combinations of the four color bounds prescribed. Many interesting problems remain open.