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

TL;DR

The paper disproves the List Square Coloring Conjecture in key regimes, notably for bipartite planar graphs and their line graphs, by constructing multiple counterexamples across line-graph families derived from prisms, generalized Petersen graphs, and bipartite planar graphs with large maximum degree. It develops a revised conjecture linking the list chromatic number to a lower bound on the chromatic number of the square, and clarifies the landscape of $i$-strong list chromatic measures with general bounds and implications for line and total graphs. Alongside, the authors provide partial results for small-order graphs via computer search and prove that many $k$-chromatic graphs with $|V|<3k$ are $1$-strongly $k$-choosable, while also constructing bounded-degree non-$k$-choosable graphs in bipartite and planar classes. The work thus sharpens the understanding of chromatic-choosability in graph squares, demonstrates the limits of existing conjectures, and offers directions for tightened, degree-sensitive formulations with potential applications to coloring problems in sparse and planar graph families.

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 $χ(G^2) \ge \frac{1}{2} Δ(G^2)+1$ (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).

Figures (20)

• Figure 1: The square of the graph $G$ is $4$-colorable (left) but not $4$-choosable (right)
• Figure 2: The square of the planar bipartite graph $P(4k, 1)$ is $4$-colorable (left) but not $4$-choosable (right)
• Figure 3: Smaller planar or biparite graphs whose square graphs are $4$-colorable but not $4$-choosable
• Figure 4: The square of the planar line graph $L(S(P(4,1)))$ is $4$-colorable (left) but not $4$-choosable (right)
• Figure 5: The square of the line graph $L(S(K_{4}))$ is $4$-colorable (left) but not $4$-choosable (right)