$2$-distance list $(\Delta +2)$-coloring of planar graphs with girth at least 10

Abstract

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say that a graph is list $k$-colorable when every vertex $v$ has a list of colors of size at least $k$. A $2$-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance list ($\Delta +2$)-coloring for planar graphs with girth at least $10$ and maximum degree $\Delta \ge 4$.

14 pages, 13 figures. arXiv admin note: substantial text overlap with arXiv:2106.03587, arXiv:2103.11687, arXiv:2105.01684, arXiv:2109.11927