Ohba’s conjecture is true for graphs K t+2,3,2∗(k−t−2),1∗t

Abstract

A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely ch(G) = χ(G). Ohba’s conjecture states that every graph G with 2χ(G)+1 or fewer vertices is chromaticchoosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. Recently, Kostochka, Stiebitz andWoodall showed that Ohba’s conjecture holds for complete multipartite graphs with partite size at most five. But the complete multipartite graphs with no restriction on their partite size, for which Ohba’s conjecture has been verified are nothing more than the graphs Kt+2,3,2∗(k−t−2),1∗t by Enotomo et al., and Kt+2,3,2∗(k−t−2),1∗t for t ≤ 4 by Shen et al.. In this paper, using the concept of f-choosable (or L0-size-choosable) of graphs, we show that Ohba’s conjecture is also true for the graphs Kt+2,3,2∗(k−t−2),1∗t when t ≥ 5. Thus, Ohba’s conjecture is true for graphs Kt+2,3,2∗(k−t−2),1∗t for all integers t ≥ 1.