Analogues of Cliques for (m,n)-colored Mixed Graphs

Julien Bensmail 1, 2 Christopher Duffy 3, 4 Sagnik Sen 3, 5
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
2 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : An (m,n)-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 (m,n)-colored mixed graph G to an (m,n)-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 (m,n)-colored mixed chromatic number, denoted chi_{m,n}(G), of an (m,n)-colored mixed graph G is the order of a smallest homomorphic image of G. An (m,n)-clique is an (m,n)-colored mixed graph C with chi_{m,n}(C) = |V(C)|. Here we study the structure of (m,n)-cliques. We show that almost all (m,n)-colored mixed graphs are (m,n)-cliques, prove bounds for the order of a largest outerplanar and planar (m,n)-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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Julien Bensmail, Christopher Duffy, Sagnik Sen. Analogues of Cliques for (m,n)-colored Mixed Graphs. Graphs and Combinatorics, Springer Verlag, 2017, 33 (4), pp.735-750. ⟨10.1007/s00373-017-1807-2⟩. ⟨hal-01078218v3⟩

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