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August, 2020 The Monochromatic Connectivity of Graphs
Zemin Jin, Xueliang Li, Kaijun Wang
Taiwanese J. Math. 24(4): 785-815 (August, 2020). DOI: 10.11650/tjm/200102

## Abstract

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\operatorname{mc}(G) \geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.

In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.

## Citation

Zemin Jin. Xueliang Li. Kaijun Wang. "The Monochromatic Connectivity of Graphs." Taiwanese J. Math. 24 (4) 785 - 815, August, 2020. https://doi.org/10.11650/tjm/200102

## Information

Received: 4 May 2018; Revised: 16 October 2018; Accepted: 7 January 2020; Published: August, 2020
First available in Project Euclid: 13 January 2020

MathSciNet: MR4124546
Digital Object Identifier: 10.11650/tjm/200102

Subjects:
Primary: 05C15, 05C40

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

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Vol.24 • No. 4 • August, 2020