Erd\H{o}s-Gallai-type results for colorful monochromatic connectivity of a graph

TL;DR

This paper investigates Erdős-Gallai-type problems related to the monochromatic connection number in edge-colored graphs, providing complete solutions to these problems and advancing understanding of graph coloring properties.

Contribution

It introduces and fully solves two Erdős-Gallai-type problems concerning the monochromatic connection number in edge-colored graphs.

Findings

Complete solutions to two Erdős-Gallai-type problems for mc(G)

Characterization of graphs with maximum monochromatic connection number

New bounds established for monochromatic connection number

Abstract

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 $G$. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erd\H{o}s-Gallai-type problems for $mc(G)$, and completely solve them.