k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prove that the $k$-forested choosability of a graph with maximum degree $\Delta\geq k\geq 4$ is at most $\lceil\frac{\Delta}{k-1}\rceil+1$, $\lceil\frac{\Delta}{k-1}\rceil+2$ or $\lceil\frac{\Delta}{k-1}\rceil+3$ if its maximum average degree is less than 12/5, $8/3 or 3, respectively.Comment: Please cite this paper in press as X. Zhang, G. Liu, J.-L. Wu, k-forested choosability of graphs with bounded maximum average degree, Bulletin of the Iranian Mathematical Society, to appea

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