PLANAR GRAPHS THAT HAVE NO SHORT CYCLES WITH A CHORD ARE 3-CHOOSABLE

Abstract

In this paper we prove that every planar graph $G$ is 3-choosable if it contains no cycle of length at most 10 with a chord. This generalizes a result obtained by Borodin [J. Graph Theory 21(1996) 183-186] and Sanders and Zhao [Graphs Combin. 11(1995) 91-94], which says that every planar graph $G$ without $k$-cycles for all $4 \leq k \leq 9$ is 3-colorable.