The List Square Coloring Conjecture fails for bipartite planar graphs and their line graphs

Abstract

Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several authors whether this conjecture holds for bipartite graphs with small degrees, claw-free graphs, or line graphs. In this paper, we introduce several kinds of counterexamples to this conjecture to solve three open problems posed by Kim and Park~(2015), Kim, Kwon, and Park~(2015), and Dai, Wang, Yang, and Yu~(2018). In particular, we disprove a planar version of this conjecture proposed by Havet, Heuvel, McDiarmid, and Reed (2017). This conjecture was originally proposed to make a stronger version of the List Total Coloring Conjecture. In order to make a revised version, it remains to decide whether this conjecture holds for bipartite graphs $G$ by imposing a lower bound on the chromatic number of the square graph $G^2$ in terms of its maximum degree as the condition $\chi(G^2) \ge \frac{1}{2} \Delta(G^2)+1$ (or by adding an upper bound on the number of colors used in lists for a weaker version). To support this version, we will show that the bipartite condition cannot be dropped even by increasing the lower bound arbitrarily. Finally, we investigate non-choosable graphs with bounded maximum degree in bipartite or planar graphs. Consequently, we improve several graph constructions due to Erd\H os, Rubin, and Taylor~(1980), Bessy, Havet, and Palaysi (2002), Voigt (1993), Mirzakhani (1996), and Glebov, Kostochka, and Tashkinov (2005) in terms of maximum degree or order. In addition, we characterize edge-minimal $3$-chromatic non-$3$-choosable (resp. $4$-chromatic non-$4$-choosable) graphs of order at most $9$ (resp. $11$) and settle a question posed by Nelsen~(2019).