Injective choosability of subcubic planar graphs with girth 6

TL;DR

This paper proves that subcubic planar graphs with girth at least 6 are injectively 5-choosable, extending previous results and highlighting the robustness of injective colorability in such graphs.

Contribution

It establishes the injective 5-choosability of subcubic planar graphs with girth 6, improving upon prior bounds and results.

Findings

Subcubic planar graphs with girth ≥ 6 are injectively 5-choosable.

Improves previous results for graphs with girth ≥ 7.

Strengthens known bounds on injective colorability.

Abstract

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lu\v{z}ar, \v{S}krekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.