A graph $G$ is $k$-choosable if it has a proper coloring for every $k$-list assignment. While every $C_3$-free planar graph is $4$-choosable, some of them are not $3$-choosable, as constructed by Voigt. Hu and Zhu conjectured that if $G$ is a $C_3$-free planar graph and $X \subseteq V(G)$ induces a bipartite subgraph, then $G$ has a proper $L$-coloring whenever $|L(x)| = 3$ for $x \in X$ and $|L(v)| = 4$ for $v \in V(G) \setminus X$. As evidence, they proved the conjecture when $X$ is an independent set. We provide further evidence by proving the conjecture when the induced subgraph $G[X]$ is an induced sparse matching. This is the first result supporting the conjecture in which the set $X$ receiving smaller lists may induce a subgraph with edges.
S. Hartke, Yu-Pei Li, Joseph Pappe et al.· 0 citations
The vertex arboricity $\mathrm{va}(G)$ of a multigraph $G$ is the minimum number $k$ for which $V(G)$ can be partitioned into $k$ subsets, each of which induces an acyclic subgraph of $G$. By definition, if $\mathrm{va}(G)= k$, then the chromatic number, $\chi(G)$, satisfies $k\leq \chi(G)\leq 2k$. Fundamental results by Borodin from 1976 and Bollob\'as and Manvel from 1979 imply an analog of Gallai's lower bound on the number of edges in a $(2k-1)$-critical graph. We consider a slight generalization of vertex arboricity in the setting of DP-coloring. Using this framework, we derive lower bounds on the number of edges in graphs critical for vertex arboricity and for list arboricity that are better than Gallai's bound, along with similar bounds in our DP-setting.
Peter Bradshaw, Alexandr V. Kostochka, Zimu Xiang· 0 citations
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