Toroidal graphs without $K_{5}^{-}$ and 6-cycles
Ping Chen, Tao Wang
TL;DR
The paper studies toroidal graphs without $K_{5}^{-}$ and $6$-cycles, aiming to tighten choosability-related bounds. Using a discharging-derived structural description (SR), it proves that such graphs are weakly $3$-degenerate, yielding DP-paint and DP-chromatic numbers at most $4$, and that their Alon-Tarsi number $AT(G)$ is at most $4$. The results are shown to be sharp in a structural sense, with two infinite families illustrating limits beyond these bounds. The work also connects to related concepts like strictly $f$-degenerate transversals and list vertex arboricity, broadening the impact on coloring theory for toroidal graphs. Overall, it strengthens known results and clarifies the landscape between list coloring, DP-coloring, and Alon-Tarsi approaches for this graph class.
Abstract
Cai et al.\ proved that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and proposed the conjecture that $\textsf{ch}(G) = 5$ if and only if $G$ contains a $K_{5}$ [J. Graph Theory 65 (2010) 1--15], where $\textsf{ch}(G)$ is the choice number of $G$. However, Choi later disproved this conjecture, and proved that toroidal graphs without $K_{5}^{-}$ (a $K_{5}$ missing one edge) and $6$-cycles are $4$-choosable [J. Graph Theory 85 (2017) 172--186]. In this paper, we provide a structural description, for toroidal graphs without $K_{5}^{-}$ and $6$-cycles. Using this structural description, we strengthen Choi's result in two ways: (I) we prove that such graphs have weak degeneracy at most three (nearly $3$-degenerate), and hence their DP-paint numbers and DP-chromatic numbers are at most four; (II) we prove that such graphs have Alon-Tarsi numbers at most $4$. Furthermore, all of our results are sharp in some sense.