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arXiv · 2103.11687

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

Hoang La·Mickael Montassier
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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.

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Hoang La, Mickael Montassier. 2021-03-22. $2$-distance $(\Delta+1)$-coloring of sparse graphs using the potential method. https://arxiv.org/abs/2103.11687

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