Analogues of Cliques for (mn)-Colored Mixed Graphs

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

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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.

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Correspondence to Sagnik Sen.

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Bensmail, J., Duffy, C. & Sen, S. Analogues of Cliques for (mn)-Colored Mixed Graphs. Graphs and Combinatorics 33, 735–750 (2017). https://doi.org/10.1007/s00373-017-1807-2

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Keywords

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