Advertisement

Graphs and Combinatorics

, Volume 33, Issue 4, pp 735–750 | Cite as

Analogues of Cliques for (mn)-Colored Mixed Graphs

  • Julien Bensmail
  • Christopher Duffy
  • Sagnik Sen
Original Paper
  • 113 Downloads

Abstract

An (mn)-colored mixed graph is a mixed graph with arcs assigned one of m different colors and edges one of n different colors. A homomorphism of an (mn)-colored mixed graph G to an (mn)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (mn)-colored mixed chromatic number, denoted \(\chi _{m,n}(G)\), of an (mn)-colored mixed graph G is the order of a smallest homomorphic image of G. An (mn)-clique is an (mn)-colored mixed graph C with \(\chi _{m,n}(C) = |V(C)|\). Here we study the structure of (mn)-cliques. We show that almost all (mn)-colored mixed graphs are (mn)-cliques, prove bounds for the order of a largest outerplanar and planar (mn)-clique and resolve an open question concerning the computational complexity of a decision problem related to (0, 2)-cliques. Additionally, we explore the relationship between \(\chi _{1,0}\) and \(\chi _{0,2}\).

Keywords

Colored mixed graphs Signed graphs Graph homomorphisms Chromatic number Clique number Planar graphs 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewer for the constructive comments towards improvement of the content, clarity and conciseness of the manuscript.

References

  1. 1.
    Alon, N., Marshall, T.H.: Homomorphisms of edge-colored graphs and Coxeter groups. J. Algebraic Comb. 8(1), 5–13 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bang-Jensen, J., Hell, P., MacGillivray, G.: The complexity of colouring by semicomplete digraphs. SIAM J. Discret. Math. 1(3), 281–298 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell). SIAM J. Comput. 38(5), 1782–1802 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bensmail, J.: On the signed chromatic number of grids. Preprint available at https://hal.archives-ouvertes.fr/hal-01349656/document (2016)
  5. 5.
    Bensmail, J., Duvignau, R., Kirgizov, S.: The complexity of deciding whether a graph admits an orientation with fixed weak diameter. Discret. Math. Theor. Comput. Sci. 17(3), 31–42 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Borodin, O.V.: On acyclic colorings of planar graphs. Discret. Math. 25(3), 211–236 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brewster, R.: Vertex colourings of edge-coloured graphs. PhD thesis, Simon Fraser University (1993)Google Scholar
  8. 8.
    Brewster, R., Foucaud, F., Hell, P., Naserasr, R.: The complexity of signed graph and 2-edge-coloured graph homomorphisms (2015). arXiv:1510.05502 (preprint)
  9. 9.
    Duffy, C.: Homomorphisms of (j, k)-mixed graphs. PhD thesis, Université de Bordeaux (2015)Google Scholar
  10. 10.
    Fertin, G., Raspaud, A., Roychowdhury, A.: On the oriented chromatic number of grids. Inf. Process. Lett. 85(5), 261–266 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goddard, W., Henning, M.A.: Domination in planar graphs with small diameter. J. Graph Theory 40, 1–25 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory Ser B 48(1), 92–110 (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Klostermeyer, W.F., MacGillivray, G.: Analogs of cliques for oriented coloring. Discuss. Math. Graph Theory 24(3), 373–388 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Klostermeyer, W.F., MacGillivray, G.: Homomorphisms and oriented colorings of equivalence classes of oriented graphs. Discret. Math. 274(1–3), 161–172 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lovász, L.: Coverings and coloring of hypergraphs. In: Proceedings of the Fourth South-eastern Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, pp. 3–12 (1973)Google Scholar
  16. 16.
    MacGillivray, G., Swarts, J.: Weak near-unanimity functions and digraph homomorphism problems. Theor. Comput. Sci. 477, 32–47 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Montejano, A., Ochem, P., Pinlou, A., Raspaud, A., Sopena, É.: Homomorphisms of 2-edge-colored graphs. Discret. Appl. Math. 158(12), 1365–1379 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nandy, A., Sen, S., Sopena, É.: Outerplanar and planar oriented cliques. J. Graph Theory 82(2), 165–193 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Naserasr, R.: Personal communication (2013)Google Scholar
  20. 20.
    Naserasr, R., Rollová, E., Sopena, É.: Homomorphisms of signed graphs. J. Graph Theory 79(3), 178–212 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nešetřil, J., Raspaud, A.: Colored homomorphisms of colored mixed graphs. J. Comb. Theory Ser. B 80(1), 147–155 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Raspaud, A., Sopena, É.: Good and semi-strong colorings of oriented planar graphs. Inf. Process. Lett. 51(4), 171–174 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sen, S.: A contribution to the theory of graph homomorphisms and colorings. PhD thesis, University of Bordeaux (2014)Google Scholar
  24. 24.
    Smolikova, P.: The simple chromatic number of oriented graphs. Electron. Notes Discret. Math. 5, 281–283 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sopena, É.: Homomorphisms and colourings of oriented graphs: an updated survey. Discret. Math. 339(7), 1993–2005 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Julien Bensmail
    • 1
  • Christopher Duffy
    • 2
  • Sagnik Sen
    • 3
  1. 1.INRIA and Université Nice-Sophia-Antipolis, I3S, UMR 7271Sophia-AntipolisFrance
  2. 2.Dalhousie UniversityHalifaxCanada
  3. 3.Ramakrishna Mission Vivekananda UniversityKolkataIndia

Personalised recommendations