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

Characterizing and recognizing exact-distance squares of graphs

“…For a graph G = (V, E) and a positive integer p, the exact distance-p graph G [ p] is the graph with vertex set V and with an edge between vertices x and y if and only if x and y have distance (exactly) p in G. In other words, G [ p] is obtained from G p by removing the edges of G p−1 . This notion has received increased attention in recent years, mostly from a graph coloring perspective [8,2,26,15,1], but also through algorithmic [25] and structural [3,9] perspectives. A result [12,22] which has sparked much of this research is the following.…”