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Chromatic Number of the Cyclic Graph of Infinite Semigroup

  • Sandeep Dalal
  • Jitender KumarEmail author
Original Paper
  • 33 Downloads

Abstract

The cyclic graph \(\Gamma (S)\) of a semigroup S is the simple graph whose vertex set is S, two element being adjacent if the subsemigroup generated by these two elements is monogenic. The purpose of this note is to prove that the chromatic number of \(\Gamma (S)\) is at most countable. The present paper generalizes the results of Shitov (Graphs Comb 33(2):485–487, 2017) and the corresponding results on power graph and enhanced power graph of groups obtained by Aalipour et al. (Electron J Comb 24(3):#P3.16, 2017).

Keywords

Monogenic semigroup Cyclic graph Power graph Enhanced power graph 

Mathematics Subject Classification

05C25 

Notes

Acknowledgements

The authors are very grateful to the referees for their valuable suggestions which lead to an improvement of this paper. The second author wishes to acknowledge the support of MATRICS Grant (MTR/2018/000779) funded by SERB.

References

  1. 1.
    Aalipour, G., Akbari, S., Cameron, P.J., Nikandish, R., Shaveisi, F.: On the structure of the power graph and the enhanced power graph of a group. Electron. J. Comb. 24(3), #P3.16 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abawajy, J., Kelarev, A., Chowdhury, M.: Power graphs: a survey. Electron. J. Graph Theory Appl. (EJGTA) 1(2), 125–147 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Afkhami, M., Jafarzadeh, A., Khashyarmanesh, K., Mohammadikhah, S.: On cyclic graphs of finite semigroups. J. Algebra Appl. 13(07), 1450035 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bera, S., Bhuniya, A.K.: On enhanced power graphs of finite groups. J. Algebra Appl. 17(8), 1850146 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chakrabarty, I., Ghosh, S., Sen, M.K.: Undirected power graphs of semigroups. Semigroup Forum 78(3), 410–426 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  7. 7.
    Kelarev, A.: Graph Algebras and Automata. Marcel Dekker, New York (2003)CrossRefGoogle Scholar
  8. 8.
    Kelarev, A., Quinn, S., Smolikova, R.: Power graphs and semigroups of matrices. Bull. Austral. Math. Soc. 63(2), 341–344 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: the influence of asymmetries. Discrete Math. 309(17), 5360–5369 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kelarev, A.V.: Ring Constructions and Applications. World Scientific, River Edge (2002)zbMATHGoogle Scholar
  11. 11.
    Kelarev, A.V.: Labelled Cayley graphs and minimal automata. Australas. J. Comb. 30, 95–101 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kelarev, A.V., Quinn, S.J.: A combinatorial property and power graphs of semigroups. Comment. Math. Univ. Carolin. 45(1), 1–7 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, C., Xu, B., Huang, H.: Cayley graphs over Green \({^*}\) relations of abundant semigroups. Graphs Comb. (2019).  https://doi.org/10.1007/s00373-019-02106-2 CrossRefGoogle Scholar
  14. 14.
    Ma, X.L., Wei, H.Q., Zhong, G.: The cyclic graph of a finite group. Algebra 2013(107265), 1–7 (2013)CrossRefGoogle Scholar
  15. 15.
    Montellano-Ballesteros, J.J., Arguello, A.S.: Hamiltonian cycles in normal Cayley graphs. Graphs Comb. (2019).  https://doi.org/10.1007/s00373-019-02090-7 CrossRefzbMATHGoogle Scholar
  16. 16.
    Shitov, Y.: Coloring the power graph of a semigroup. Graphs Comb. 33(2), 485–487 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, New Jersey (1996)zbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBirla Institute of Technology and Science PilaniPilaniIndia

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