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

TL;DR

This paper proves that planar graphs with girth at least 10 and maximum degree at least 4 can be properly colored with a list of (+2) colors such that no two vertices within distance two share the same color.

Contribution

It establishes the existence of a 2-distance list ((+2))-coloring for a specific class of planar graphs, extending previous coloring results.

Findings

Validates the coloring for graphs with girth  at least 10

Applicable to graphs with maximum degree  

Advances understanding of list colorings in planar graphs

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 $\Delta\geq 4$.