$2$-distance, injective, and exact square list-coloring of planar graphs with maximum degree 4

TL;DR

This paper investigates various distance-based list-colorings of planar graphs with maximum degree 4, establishing new bounds for 2-distance, injective, and exact square colorings under certain girth conditions.

Contribution

It provides new bounds for 2-distance, injective, and exact square list-colorings of planar graphs with maximum degree 4, extending previous results in the field.

Findings

Planar graphs with max degree 4 and girth at least 4 are 2-distance list (Δ+7)-colorable.

Planar graphs with max degree 4 are injectively list (Δ+7)-colorable.

Planar graphs with max degree 4 are exact square list (Δ+6)-colorable.

Abstract

In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graphs with maximum degree $\Delta = 4$ and girth at least $4$ are $2$-distance list $(\Delta + 7)$-colorable and injectively list $(\Delta + 5)$-colorable. Additionally, we prove that planar graphs with $\Delta = 4$ are injectively list $(\Delta + 7)$-colorable and exact square list $(\Delta + 6)$-colorable.