On incidence coloring conjecture in Cartesian products of graphs

TL;DR

This paper investigates the incidence coloring conjecture in Cartesian product graphs, identifying conditions on factor graphs that ensure the conjecture holds, thus advancing understanding of incidence colorings in complex graph structures.

Contribution

It introduces sufficient properties of factor graphs in Cartesian products that guarantee incidence colorings with at most 1 colors, extending known results.

Findings

Identifies properties of factor graphs that ensure the conjecture holds.

Provides conditions under which Cartesian product graphs admit optimal incidence colorings.

Advances the theoretical understanding of incidence coloring in product graphs.

Abstract

An incidence in a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $v$. Two incidences $(v,e)$ and $(u,f)$ are adjacent if at least one of the following holds: $(a)$ $v = u$, $(b)$ $e = f$, or $(c)$ $vu \in \{e,f\}$. An incidence coloring of $G$ is a coloring of its incidences assigning distinct colors to adjacent incidences. It was conjectured that at most $\Delta(G) + 2$ colors are needed for an incidence coloring of any graph $G$. The conjecture is false in general, but the bound holds for many classes of graphs. We introduce some sufficient properties of the two factor graphs of a Cartesian product graph $G$ for which $G$ admits an incidence coloring with at most $\Delta(G) + 2$ colors.