## Taiwanese Journal of Mathematics

### The Monochromatic Connectivity of Graphs

#### Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 13 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1578884422

Digital Object Identifier
doi:10.11650/tjm/200102

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

#### Citation

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

#### References

• B. Bollobás, Extremal Graph Theory, London Mathematical Society Monographs 11, Academic Press, New York, 1978.
• Q. Cai, X. Li and D. Wu, Erdős-Gallai-type results for colorful monochromatic connectivity of a graph, J. Comb. Optim. 33 (2017), no. 1, 123–131.
• Y. Caro and R. Yuster, Colorful monochromatic connectivity, Discrete Math. 311 (2011), no. 16, 1786–1792.
• G. Chartrand, G. L. Johns, K. A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008), no. 1, 85–98.
• R. Gu, X. Li, Z. Qin and Y. Zhao, More on the colorful monochromatic connectivity, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 4, 1769–1779.
• X. Li and Y. Sun, Rainbow Connections of Graphs, SpringerBriefs in Mathematics, Springer, New York, 2012.
• X. Li and D. Wu, The (vertex-)monochromatic index of a graph, J. Comb. Optim. 33 (2017), no. 4, 1443–1453.