Taiwanese Journal of Mathematics

The Monochromatic Connectivity of Graphs

Zemin Jin, Xueliang Li, and Kaijun Wang

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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.

Article information

Taiwanese J. Math., Advance publication (2020), 31 pages.

First available in Project Euclid: 13 January 2020

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Digital Object Identifier

Primary: 05C15: Coloring of graphs and hypergraphs 05C40: Connectivity

monochromatic path monochromatic connectivity MC-coloring


Jin, Zemin; Li, Xueliang; Wang, Kaijun. The Monochromatic Connectivity of Graphs. Taiwanese J. Math., advance publication, 13 January 2020. doi:10.11650/tjm/200102. https://projecteuclid.org/euclid.twjm/1578884422

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