List coloring $C_3$-free planar graphs with a sparse matching of restricted lists

Abstract

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.