On clique numbers of colored mixed graphs

An (m, n)-colored mixed graph, or simply, an (m, n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m, n)-graph G to another (m, n)-graph H is a vertex mapping that preserves adjacency; and the type and direction of the adjacency. An (m, n)-relative clique of G is a vertex subset R whose images are always distinct under any homomorphism of G to any H. The maximum cardinality of an (m, n)-relative clique of a graph is called the (m, n)-relative clique number of the graph. In this article, we explore the (m, n)-relative clique numbers for three different families of graphs, namely, graphs having bounded maximum degree increment , subcubic graphs, partial 2-trees and planar graphs and provide tight or close bounds in most cases.(c) 2022 Elsevier B.V. All rights reserved.