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# Mathematics > Combinatorics

# Title: $2$-distance $(Δ+2)$-coloring of sparse graphs

(Submitted on 24 Sep 2021)

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 ($\Delta+2$)-coloring for graphs with maximum average degree less than $\frac{8}{3}$ (resp. $\frac{14}{5}$) and maximum degree $\Delta\geq 6$ (resp. $\Delta\geq 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $\Delta\geq 6$ (resp. $\Delta\geq 10$) admits a $2$-distance $(\Delta+2)$-coloring.

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