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.

DOI

10.1016/j.disc.2011.05.017

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