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).