Erd\H{o}s-Gallai-type results for total monochromatic connection of graphs

TL;DR

This paper investigates Erdős-Gallai-type problems related to the total monochromatic connection number in total-colored graphs, providing complete solutions for two variants of these problems.

Contribution

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

Findings

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

Characterization of graphs based on total monochromatic connection number

New bounds and properties for total-colored graphs

Abstract

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. In this paper, we study two kinds of Erd\H{o}s-Gallai-type problems for $tmc(G)$ and completely solve them.