Monochromatic connectivity and graph products

TL;DR

This paper investigates the monochromatic connection number in various graph products, providing bounds and insights into how this parameter behaves under different graph operations.

Contribution

It introduces bounds for the monochromatic connection number in lexicographical, strong, Cartesian, and direct graph products, expanding understanding of this parameter in complex graph structures.

Findings

Established upper and lower bounds for the monochromatic connection number in graph products.

Analyzed how graph products affect monochromatic connectivity properties.

Extended previous concepts to new classes of graph products.

Abstract

The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} ($MC$-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an $MC$-coloring of a graph $G$. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.