# More on the Colorful Monochromatic Connectivity

• Ran Gu
• Xueliang Li
• Zhongmei Qin
• Yan Zhao
Article

## Abstract

An edge-coloring of a connected graph is a monochromatically-connecting coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices, which was introduced by Caro and Yuster. Let mc(G) denote the maximum number of colors used in an MC-coloring of a graph G. Note that an MC-coloring does not exist if G is not connected, in which case we simply let $$mc(G)=0$$. In this paper, we characterize all connected graphs of size m with $$mc(G)=1, 2, 3, 4$$, $$m-1$$, $$m-2$$ and $$m-3$$, respectively. We use G(np) to denote the Erdős-Rényi random graph model, in which each of the $$\left( {\begin{array}{c}n\\ 2\end{array}}\right)$$ pairs of vertices appears as an edge with probability p independent from other pairs. For any function f(n) satisfying $$1\le f(n)<\frac{1}{2}n(n-1)$$, we show that if $$\ell n \log n\le f(n)<\frac{1}{2}n(n-1)$$, where $$\ell \in \mathbb {R}^+$$, then $$p=\frac{f(n)+n\log \log n}{n^2}$$ is a sharp threshold function for the property $$mc\left( G\left( n,p\right) \right) \ge f(n)$$; if $$f(n)=o(n\log n)$$, then $$p=\frac{\log n}{n}$$ is a sharp threshold function for the property $$mc\left( G\left( n,p\right) \right) \ge f(n)$$.

## Keywords

Coloring Monochromatic Connectivity Random graphs

## Mathematics Subject Classification

05C15 05C40 05C80

## Notes

### Acknowledgments

This worl was supported by NSFC Nos. 11371205 and 11531011, and PCSIRT. The authors are very grateful to the reviewers for their valuable suggestions and comments, which helped to improve the presentation of the paper.

## References

1. 1.
Alon, N., Spencer, J.: The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd edn. Wiley, Hoboken (2008)
2. 2.
Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, Berlin (2008)
3. 3.
Bollobás, B., Thomason, A.: Threshold functions. Combinatorica 7, 35–38 (1986)
4. 4.
Cai, Q., Li, X., Wu, D.: Erdős-Gallai-type results for colorful monochromatic connectivity of a graph. J. Comb. Optim. doi:
5. 5.
Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: On rainbow connection. Electron. J. Comb. 15, #R57 (2008)
6. 6.
Caro, Y., Yuster, R.: Colorful monochromatic connectivity. Discrete Math. 311, 1786–1792 (2011)
7. 7.
Chen, L., Li, X., Yang, K., Zhao, Y.: The 3-rainbow index of a graph. Discuss. Math. Graph Theory 35, 81–94 (2015)
8. 8.
Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133, 85–98 (2008)
9. 9.
Chartrand, G., Johns, G., McKeon, K., Zhang, P.: The rainbow connectivity of a graph. Networks 54(2), 75–81 (2009)
10. 10.
Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)
11. 11.
Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Am. Math. Soc. 124, 2993–3002 (1996)
12. 12.
He, J., Liang, H.: On rainbow-$$k$$-connectivity of random graphs. Inf. Process. Lett. 112, 406–410 (2012)
13. 13.
Huang, X., Li, X., Shi, Y.: Note on the hardness of rainbow connections for planar and line graphs. Bull. Malays. Math. Sci. Soc. 38, 1235–1241 (2015)
14. 14.
Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: a survey. Graph Combin. 29, 1–38 (2013)
15. 15.
Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Briefs in Mathematics. Springer, New York (2012)
16. 16.
Li, X., Sun, Y., Zhao, Y.: Characterization of graphs with rainbow connection number $$m-2$$ and $$m-3$$. Aust. J. Comb. 60, 306–313 (2014)
17. 17.
Li, X., Schiermeyer, I., Yang, K., Zhao, Y.: Graphs with 3-rainbow index $$n-1$$ and $$n-2$$. Discuss. Math. Graph Theory 35(1), 105–120 (2015)
18. 18.
Li, X., Schiermeyer, I., Yang, K., Zhao, Y.: Graphs with 4-rainbow index 3 and $$n-1$$. Discuss. Math. Graph Theory 35(2), 387–398 (2015)