Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. 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 Gwithmaximumdegree ∆(G) satisfies lcl(G) ≥ ⌈∆(G)/2⌉ + 1. In this paper, we prove the following results: (1) if mad(G) < 12/5 and∆(G) ≥ 3, then lcl(G) = ⌈∆(G)/2⌉+1, andwe give an infinite family of examples to show that this result is best possible; (2) if mad(G) < 3 and∆(G) ≥ 9, then lcl(G) ≤ ⌈∆(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) ≤ ⌈∆(G)/2⌉ + 4. © 2011 Elsevier B.V. All rights reserved.