Star Coloring of the Cartesian Product of Cycles

TL;DR

This paper proves that the Cartesian product of most pairs of cycles can be properly star-colored with five colors, except for two specific cases, expanding understanding of star colorings in graph products.

Contribution

It establishes that all Cartesian products of two cycles, excluding two specific cases, admit a 5-star coloring, providing new insights into graph coloring properties.

Findings

Most cycle Cartesian products have a 5-star coloring.

Exceptions are the products C3×C3 and C3×C5.

The result broadens understanding of star colorings in graph theory.

Abstract

A proper vertex coloring of a graph $G$ is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored $P_4$ in $G$. In this paper, we show that the Cartesian product of any two cycles, except $C_3 \square C_3$ and $C_3 \square C_5$, has a $5$-star coloring.