Graphs and Combinatorics

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

Analogues of Cliques for (mn)-Colored Mixed Graphs

Original Paper


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 \(\chi _{0,2}\).


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


  1. 1.
    Alon, N., Marshall, T.H.: Homomorphisms of edge-colored graphs and Coxeter groups. J. Algebraic Comb. 8(1), 5–13 (1998)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bensmail, J.: On the signed chromatic number of grids. Preprint available at (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)MathSciNetMATHGoogle Scholar
  6. 6.
    Borodin, O.V.: On acyclic colorings of planar graphs. Discret. Math. 25(3), 211–236 (1979)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goddard, W., Henning, M.A.: Domination in planar graphs with small diameter. J. Graph Theory 40, 1–25 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory Ser B 48(1), 92–110 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Klostermeyer, W.F., MacGillivray, G.: Analogs of cliques for oriented coloring. Discuss. Math. Graph Theory 24(3), 373–388 (2004)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nandy, A., Sen, S., Sopena, É.: Outerplanar and planar oriented cliques. J. Graph Theory 82(2), 165–193 (2016)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nešetřil, J., Raspaud, A.: Colored homomorphisms of colored mixed graphs. J. Comb. Theory Ser. B 80(1), 147–155 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Raspaud, A., Sopena, É.: Good and semi-strong colorings of oriented planar graphs. Inf. Process. Lett. 51(4), 171–174 (1994)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sopena, É.: Homomorphisms and colourings of oriented graphs: an updated survey. Discret. Math. 339(7), 1993–2005 (2016)MathSciNetCrossRefMATHGoogle 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

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