More on the Colorful Monochromatic Connectivity

Article
  • 102 Downloads

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)CrossRefMATHGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Bollobás, B., Thomason, A.: Threshold functions. Combinatorica 7, 35–38 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cai, Q., Li, X., Wu, D.: Erdős-Gallai-type results for colorful monochromatic connectivity of a graph. J. Comb. Optim. doi:  10.1007/s10878-015-9938-y
  5. 5.
    Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: On rainbow connection. Electron. J. Comb. 15, #R57 (2008)MathSciNetMATHGoogle Scholar
  6. 6.
    Caro, Y., Yuster, R.: Colorful monochromatic connectivity. Discrete Math. 311, 1786–1792 (2011)MathSciNetCrossRefMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133, 85–98 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Chartrand, G., Johns, G., McKeon, K., Zhang, P.: The rainbow connectivity of a graph. Networks 54(2), 75–81 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)MathSciNetMATHGoogle Scholar
  11. 11.
    Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Am. Math. Soc. 124, 2993–3002 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    He, J., Liang, H.: On rainbow-\(k\)-connectivity of random graphs. Inf. Process. Lett. 112, 406–410 (2012)MathSciNetCrossRefMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: a survey. Graph Combin. 29, 1–38 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Briefs in Mathematics. Springer, New York (2012)CrossRefMATHGoogle Scholar
  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)MathSciNetMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Center for Combinatorics and LPMC-TJKLCNankai UniversityTianjinChina

Personalised recommendations