Bad list assignments for non‐k $k$‐choosable k $k$‐chromatic graphs with 2k+2 $2k+2$‐vertices

Abstract

AbstractIt was conjectured by Ohba, and proved by Noel, Reed and Wu that ‐chromatic graphs with are chromatic‐choosable. This upper bound on is tight: if is even, then and are ‐chromatic graphs with vertices that are not chromatic‐choosable. It was proved by Zhu and Zhu that these are the only non‐‐choosable complete ‐partite graphs with vertices. For , a bad list assignment of is a ‐list assignment of such that is not ‐colourable. Bad list assignments for were characterized by Enomoto, Ohba, Ota and Sakamoto. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for . Using these results, we characterize all non‐‐choosable ‐chromatic graphs with vertices.