On clique numbers of colored mixed graphs

TL;DR

This paper investigates the maximum size of special vertex subsets called (m,n)-relative cliques in various colored mixed graphs, which are graphs with multiple arc and edge types, and their implications for graph homomorphisms.

Contribution

It introduces the concept of (m,n)-relative clique numbers and explores their values across different graph families, advancing understanding of graph homomorphisms with multiple edge types.

Findings

Defined (m,n)-relative clique numbers for various graph families

Established bounds and exact values for specific graph classes

Enhanced understanding of homomorphism-preserving vertex subsets

Abstract

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, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. 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 various families of graphs.