Exploring sigma coloring within lexicographic products of innovative graph families

Abstract

Sigma coloring is a vertex coloring strategy that distinguishes adjacent vertices by the sum of colors in their open neighborhood. The minimum number of colors needed in a sigma coloring of a graph $G$ is called its sigma chromatic number $\sigma\left(G\right)$. The bounds of $\sigma\left(G\right)$ are sharp. For every graph $G$, $\sigma\left(G\right)\leq\chi\left(G\right)$. In this article, a clear investigation is made under sigma coloring on the lexicographic product of a path graph with various graph families including path, cycle, star, double star, and comb graphs. Additionally, the sigma chromatic number for each case is found and verified with chromatic number for bounds.