#### Title

Linear choosability of sparse graphs

#### Document Type

Article

#### Department/Program

Mathematics

#### Journal Title

Discrete Mathematics

#### Pub Date

2011

#### Volume

311

#### Issue

17

#### First Page

1910

#### Abstract

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 Ic(e)(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 Ice(G) > = inverted right perpendicular Delta(G)/2inverted left perpendicular + 1. In this paper, we prove the following results: (1) if mad(G) < 12/5 and Delta(G) > = 3, then Ic(e)(G) = inverted right perpendicular Delta(G)/2inverted left perpendicular + 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) > = 9, then Ic(e) (G) < = inverted right perpendicular Delta(G)/2inverted left perpendicular + 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 Ic(e)(G) < = inverted right perpendicular Delta(G)/2inverted left perpendicular + 4. (C) 2011 Elsevier B.V. All rights reserved.

#### Recommended Citation

Cranston, Daniel W. and Yu, Gexin, Linear choosability of sparse graphs (2011). *Discrete Mathematics*, 311(17), 1910-1917.

10.1016/j.disc.2011.05.017

#### DOI

10.1016/j.disc.2011.05.017