Skip to main content
SpringerLink
Log in

Distance labellings of Cayley graphs of semigroups

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups S and their subsets C such that \(\,{\mathrm {Cay}}(S,C)\) is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of \(\,{\mathrm {Cay}}(S,C)\). We then describe all graphs with minimum spans satisfying the same restrictions, and give examples to show that a fairly straightforward upper bound for the minimum spans of the underlying undirected graphs of Cayley graphs turns out to be sharp even for the class of combinatorial semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abawajy, J., Kelarev, A.V.: Classification systems based on combinatorial semigroups. Semigroup Forum 86, 603–612 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon, N., Mohar, B.: The chromatic number of graph powers. Comb. Probab. Comput. 11, 1–10 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brankovic, L., Miller, M., Plesnik, J., Ryan, J., Siran, J.: A note on constructing large Cayley graphs of given degree and diameter by voltage assignment. Electron. J. Comb. 5, 147–158 (1998)

    MathSciNet  Google Scholar 

  4. Calamoneri, T.: The \(L(h, k)\)-labelling problem: a survey and annotated bibliography. Comput. J. 49, 585–608 (2006)

    Article  Google Scholar 

  5. Chang, G.J., Lu, C., Zhou, S.: No-hole 2-distant colorings for Cayley graphs on finitely generated abelian groups. Discret. Math. 307, 1808–1817 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chang, G.J., Lu, C., Zhou, S.: Distance-two labellings of Hamming graphs. Discret. Appl. Math. 157, 1896–1904 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dénes, J.: Connections between transformation semigroups and graphs. In: Theory of Graphs, Gordon and Breach, New York, pp. 93–101 (1967)

  8. Fan, S., Zeng, Y.: On Cayley graphs of bands. Semigroup Forum 74, 99–105 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gao, X., Liu, W., Luo, Y.: On Cayley graphs of normal bands. Ars Comb. 100, 409–419 (2011)

    MATH  MathSciNet  Google Scholar 

  10. Hao, Y., Gao, X., Luo, Y.: On the Cayley graphs of Brandt semigroups. Commun. Algebra 39, 2874–2883 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Howie, J.M.: Fundamentals Semigroup Theory. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  12. Jiang, Z.: An answer to a question of Kelarev and Praeger on Cayley graphs of semigroups. Semigroup Forum 69, 457–461 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Kelarev, A.V.: On undirected Cayley graphs. Australas. J. Comb. 25, 73–78 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Kelarev, A.V.: Labelled Cayley graphs and minimal automata. Australas. J. Comb. 30, 95–101 (2004)

    MATH  MathSciNet  Google Scholar 

  15. Kelarev, A.V.: On Cayley graphs of inverse semigroups. Semigroup Forum 72, 411–418 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kelarev, A.V.: Graph Algebras and Automata. Marcel Dekker, New York (2003)

    MATH  Google Scholar 

  17. Kelarev, A.V., Praeger, C.E.: On transitive Cayley graphs of groups and semigroups. Eur. J. Comb. 24, 59–72 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kelarev, A.V., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: the influence of asymmetries. Discret. Math. 309, 5360–5369 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kelarev, A.V., Yearwood, J.L., Watters, P.A.: Optimization of classifiers for data mining based on combinatorial semigroups. Semigroup Forum 82, 242–251 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Khosravi, B., Khosravi, B.: A characterization of Cayley graphs of Brandt semigroups. Bull. Malays. Math. Sci. Soc. 35, 399–410 (2012)

    MATH  MathSciNet  Google Scholar 

  21. Khosravi, B., Khosravi, B.: On Cayley graphs of semilattices of semigroups. Semigroup Forum 86, 114–132 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Khosravi, B., Khosravi, B.: On combinatorial properties of bands. Commun. Algebra 42, 1379–1395 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  23. Khosravi, B., Khosravi, B., Khosravi, B.: On color-automorphism vertex transitivity of semigroups. Eur. J. Comb. 40, 55–64 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  24. Khosravi, B., Khosravi, B., Khosravi, B.: On the Cayley D-saturated property of semigroups. Semigroup Forum (2015). doi:10.1007/s00233-015-9716-2

  25. Khosravi, B., Mahmoudi, M.: On Cayley graphs of rectangular groups. Discret. Math. 310, 804–811 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. King, D., Ras, C.J., Zhou, S.: The \(L(h, 1, 1)\)-labelling problem for trees. Eur. J. Comb. 31, 1295–1306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. King, D., Li, K.Y., Zhou, S.: Linear and cyclic distance-three labellings of trees. Discret. Appl. Math. 178, 109–120 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  28. Knauer, U.: Algebraic Graph Theory. Morphisms, Monoids and Matrices. Walter de Gruyter & Co., Berlin (2011)

    Book  MATH  Google Scholar 

  29. Lakshmivarahan, S., Jwo, J.-S., Dhall, S.K.: Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey. Parallel Comput. 19, 361–407 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Li, C.H., Praeger, C.E.: On the isomorphism problem for finite Cayley graphs of bounded valency. Eur. J. Comb. 20, 279–292 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Luo, Y., Hao, Y., Clarke, G.T.: On the Cayley graphs of completely simple semigroups. Semigroup Forum 82, 288–295 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Panma, S., Chiangmai, N., Knauer, U., Arworn, Sr: Characterizations of Clifford semigroup digraphs. Discret. Math. 306, 1247–1252 (2006)

    Article  MATH  Google Scholar 

  33. Praeger, C.E.: Finite normal edge-transitive Cayley graphs. Bull. Aust. Math. Soc. 60, 207–220 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang, S., Li, Y.: On Cayley graphs of completely 0-simple semigroups. Cent. Eur. J. Math. 11, 924–930 (2013)

    MATH  MathSciNet  Google Scholar 

  35. Zelinka, B.: Graphs of semigroups. Casopis. Pest. Mat. 106, 407–408 (1981)

    MATH  MathSciNet  Google Scholar 

  36. Zhou, S.: Labelling Cayley graphs on abelian groups. SIAM J. Discret. Math. 19, 985–1003 (2006)

    Article  MATH  Google Scholar 

  37. Zhou, S.: A distance labelling problem for hypercubes. Discret. Appl. Math. 156, 2846–2854 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Andrei Kelarev, Charl Ras & Sanming Zhou

Authors
  1. Andrei Kelarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charl Ras
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sanming Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charl Ras.

Additional information

Communicated by Marcel Jackson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kelarev, A., Ras, C. & Zhou, S. Distance labellings of Cayley graphs of semigroups. Semigroup Forum 91, 611–624 (2015). https://doi.org/10.1007/s00233-015-9748-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-015-9748-7

Keywords

Search

Navigation