Exact square coloring of subcubic planar graphs

TL;DR

This paper investigates the exact square chromatic number of subcubic planar graphs, establishing tight bounds for certain subclasses and characterizing fullerene graphs with specific coloring properties.

Contribution

It provides new bounds and characterizations for the exact square chromatic number in subcubic planar graphs and fullerene graphs, including computational verification.

Findings

Subcubic bipartite planar graphs have an exact square chromatic number at most 4.

Subcubic K4-minor-free graphs have an exact square chromatic number at most 4.

Fullerene graphs with certain properties have an exact square chromatic number of 3 or at most 5.

Abstract

We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $\chi^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite planar graphs and subcubic K 4-minor-free graphs have exact square chromatic number at most 4. We then turn our attention to the class of fullerene graphs, which are cubic planar graphs with face sizes 5 and 6. We characterize fullerene graphs with exact square chromatic number 3. Furthermore, supporting a conjecture of Chen, Hahn, Raspaud and Wang (that all subcubic planar graphs are injectively 5-colorable) we prove that any induced subgraph of a fullerene graph has exact square chromatic number at most 5. This is done by first proving that a minimum counterexample has to be on at most 80 vertices and then computationally verifying the claim for all such graphs.