$2$-distance $(\Delta+1)$-coloring of sparse graphs using the potential method

TL;DR

This paper proves that sparse graphs with certain degree constraints can be properly colored with $ ext{max degree}+1$ colors using the potential method, improving understanding of 2-distance coloring in sparse graph classes.

Contribution

It introduces a novel application of the potential method to establish 2-distance $( ext{max degree}+1)$-colorings for graphs with bounded average degree and planar graphs with high girth.

Findings

Graphs with max average degree less than 18/7 and max degree at least 7 are 2-distance $( ext{max degree}+1)$-colorable.

Planar graphs with girth at least 9 and max degree at least 7 admit such colorings.

The potential method effectively reduces configurations compared to classic approaches.

Abstract

A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+1$)-coloring for graphs with maximum average degree less than $\frac{18}{7}$ and maximum degree $\Delta\geq 7$. As a corollary, every planar graph with girth at least $9$ and $\Delta\geq 7$ admits a $2$-distance $(\Delta+1)$-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on $2$-distance coloring.