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

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 Δ=4 and girth at least 4 are 2-distance list (Δ+7)-colorable and injectively list (Δ+5)-colorable. Additionally, we prove that planar graphs with Δ=4 are injectively list (Δ+7)-colorable and exact square list (Δ+6)-colorable.