Proper conflict-free choosability of sparse graphs with girth at least seven

Abstract

A proper conflict-free coloring of a graph is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its neighborhood. A graph $G$ is proper conflict-free $(\mathrm{degree}+2)$-choosable if every list assignment $L$ with $|L(v)|\ge d_G(v)+2$ for each $v\in V(G)$ admits such a coloring from the lists. We prove that every graph with girth at least $7$ and maximum average degree less than $8/3$ is proper conflict-free $(\mathrm{degree}+2)$-choosable. Consequently, every planar graph of girth at least $8$ has this property, improving the previously established sufficient girth bound of $9$.

arXiv comment14 pages; revised exposition and clarified proofs; references updated; main results unchanged