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 SenEmail author
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 



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


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Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Julien Bensmail
    • 1
  • Christopher Duffy
    • 2
  • Sagnik Sen
    • 3
    Email author
  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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