Star Coloring of Tensor Product of Two Graphs

TL;DR

This paper investigates the star coloring properties of tensor product graphs, establishing bounds, exact values for specific cases, and identifying exceptions, thereby advancing understanding of coloring complexities in graph tensor products.

Contribution

It provides new bounds and exact values for the star chromatic number of tensor products of graphs, including paths and cycles, with specific exceptions.

Findings

Star chromatic number of tensor product of two paths is exactly determined.

Star chromatic number of tensor product of two cycles is five, with specific exceptions.

Tight bounds are established for cycle-path tensor products.

Abstract

A star coloring of a graph $G$ is a proper vertex coloring such that no path on four vertices is bicolored. The smallest integer $k$ for which $G$ admits a star coloring with $k$ colors is called the star chromatic number of $G$, denoted as $\chi_s(G)$. In this paper, we study the star coloring of tensor product of two graphs and obtain the following results. 1. We give an upper bound on the star chromatic number of the tensor product of two arbitrary graphs. 2. We determine the exact value of the star chromatic number of tensor product two paths. 3. We show that the star chromatic number of tensor product of two cycles is five, except for $C_3 \times C_3$ and $C_3 \times C_5$. 4. We give tight bounds for the star chromatic number of tensor product of a cycle and a path.