Strong edge-colorings for k-degenerate graphs

We prove that the strong chromatic index for each $k$-degenerate graph with maximum degree $\Delta$ is at most $(4k-2)\Delta-k(2k-1)+1$

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