Journal of Combinatorial Optimization

, Volume 35, Issue 4, pp 1300–1311 | Cite as

Some extremal results on the colorful monochromatic vertex-connectivity of a graph

Article
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Abstract

A path in a vertex-colored graph is called a vertex-monochromatic path if its internal vertices have the same color. A vertex-coloring of a graph is a monochromatic vertex-connection coloring (MVC-coloring for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph G, the monochromatic vertex-connection number, denoted by mvc(G), is defined to be the maximum number of colors used in an MVC-coloring of G. These concepts of vertex-version are natural generalizations of the colorful monochromatic connectivity of edge-version, introduced by Caro and Yuster (Discrete Math 311:1786–1792, 2011). In this paper, we mainly investigate the Erdős–Gallai-type problems for the monochromatic vertex-connection number mvc(G) and completely determine the exact value. Moreover, the Nordhaus–Gaddum-type inequality for mvc(G) is also given.

Keywords

Vertex-monochromatic path MVC-coloring Monochromatic vertex-connection number Erdős–Gallai-type problem Nordhaus–Gaddum-type problem 

Mathematics Subject Classification

05C15 05C35 05C38 05C40 

Notes

Acknowledgements

The authors are very grateful to the reviewers for their useful comments and suggestions, which helped to improve the presentation of the paper.

References

  1. Bondy JA, Murty USR (2008) Graph theory, GTM 244. Springer, BerlinCrossRefMATHGoogle Scholar
  2. Cai Q, Li X, Wu D (2017) Erdős–Gallai-type results for colorful monochromatic connectivity of a graph. J Comb Optim 33(1):123–131MathSciNetCrossRefMATHGoogle Scholar
  3. Caro Y, West DB, Yuster R (2000) Connected domination and spanning trees with many leaves. SIAM J Discrete Math 13(2):202–211MathSciNetCrossRefMATHGoogle Scholar
  4. Caro Y, Yuster R (2011) Colorful monochromatic connectivity. Discrete Math 311:1786–1792MathSciNetCrossRefMATHGoogle Scholar
  5. Chen L, Li X, Liu M (2011) Nordhaus–Gaddum-type bounds for rainbow vertex-connection number of a graph. Utilitas Math 86:335–340MathSciNetMATHGoogle Scholar
  6. Chen L, Li X, Lian H (2013) Nordhaus–Gaddum-type theorem for rainbow connection number of graphs. Gr Comb 29(5):1235–1247MathSciNetCrossRefMATHGoogle Scholar
  7. Ding G, Johnson T, Seymour P (2001) Spanning trees with many leaves. J Gr Theory 37:189–197MathSciNetCrossRefMATHGoogle Scholar
  8. Griggs JR, Wu M (1992) Spanning trees in graphs of minimum degree 4 or 5. Discrete Math 104(2):167–183MathSciNetCrossRefMATHGoogle Scholar
  9. Harary F (1962) The maximum connnectivity of a graph. Mathematics 48:1142–1146MATHGoogle Scholar
  10. Harary F, Haynes TW (1996) Nordhaus–Gaddum inequalities for domination in graphs. Discrete Math 155:99–105MathSciNetCrossRefMATHGoogle Scholar
  11. Harary F, Robinson RW (1985) The diameter of a graph and its complement. Am Math Mon 92:211–212MathSciNetCrossRefMATHGoogle Scholar
  12. Kemnitz A, Schiermeyer I (2011) Graphs with rainbow connection number two. Discuss Math Gr Theory 31:313–320MathSciNetCrossRefMATHGoogle Scholar
  13. Kleitman DJ, West DB (1991) Spanning trees with many leaves. SIAM J Discrete Math 4(1):99–106MathSciNetCrossRefMATHGoogle Scholar
  14. Li H, Li X, Sun Y, Zhao Y (2014) Note on minimally d-rainbow connected graphs. Gr Comb 30(4):949–955MathSciNetCrossRefMATHGoogle Scholar
  15. Li X, Liu M, Schiermeyer I (2013) Rainbow connection number of dense graphs. Discuss Math Gr Theory 33:603–611MathSciNetCrossRefMATHGoogle Scholar
  16. Li X, Mao Y (2015) Nordhaus–Gaddum-type results for the generalized edge-connectivity of graphs. Discrete Appl Math 185:102–112MathSciNetCrossRefMATHGoogle Scholar
  17. Li X, Shi Y, Sun Y (2013) Rainbow connections of graphs: a survey. Gr Comb 29:1–38MathSciNetCrossRefMATHGoogle Scholar
  18. Li X, Sun Y (2012) Rainbow connections of graphs. Springer briefs in mathematics. Springer, New YorkCrossRefGoogle Scholar
  19. Li X, Wu D (2018) A survey on monochromatic connections of graphs. Theory Appl Gr 4:1–21Google Scholar
  20. Lo A (2014) A note on the minimum size of \(k\)-rainbow-connected graphs. Discrete Math 331:20–21MathSciNetCrossRefMATHGoogle Scholar
  21. Nordhaus EA, Gaddum JW (1956) On complementary graphs. Am Math Mon 63:175–177CrossRefMATHGoogle Scholar
  22. Schiermeyer I (2013) On minimally rainbow \(k\)-connected graphs. Discrete Appl Math 161:702–705MathSciNetCrossRefMATHGoogle Scholar
  23. Zhang L, Wu B (2005) The Nordhaus–Gaddum-type inequalities of some chemical indices. MATCH Commun Math Comput Chem 54:189–194MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Combinatorics and LPMCNankai UniversityTianjinChina

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