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

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 χ 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 χ 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 χ 1,0 and χ 0,2 .

Oriented colourings of graphs with maximum degree three and four

“…It is therefore easy to check whether a given oriented graph is an oclique. On the other hand, it is NP-complete to decide whether a given graph can be oriented to be an oclique [1].…”

“…We claim the vertices of G − z are configured as in one of the two partially oriented graphs in Figure 9 where the unoriented edges are oriented so that each of A v 2 and A v 2 contains an element from Z. By the previous argument, u 1 and v 1 (respectively u 2 and v 2 ) have a pair of common neighbours. Since A v 1 (respectively A v 2 ) contains a copy of Z ∈ Z (respectively Z ∈ Z), these common neighbours must either be adjacent or are the ends of a 2-dipath.…”

“…. , k − 1} such that Condition (1) requires that adjacent vertices are assigned different colours. Condition (2) can be viewed as the requirement that the colour assignment takes the orientation into account: if there is an edge from a vertex of colour i to a vertex of colour j, then there is no edge from a vertex of colour j to a vertex of colour i.…”

Relative clique number of planar signed graphs

“…For simple undirected graphs, the absolute and relative clique numbers coincide which is not the case when (m, n) = (0, 1) [2].…”

“…That is, no two distinct vertices of a relative clique can be identified under any homomorphism. The (m, n)-relative clique number ω r(m,n) (G) of an (m, n)-colored mixed graph G is the cardinality of a largest relative (m, n)-clique of G. Bensmail, Duffy and Sen [2] showed that ω a(m,n) (G) ≤ ω r(m,n) (G) ≤ χ m,n (G).…”