On relative clique number of colored mixed graphs

TL;DR

This paper investigates the maximum size of vertex sets in colored mixed graphs that must remain distinct under any homomorphism, focusing on specific graph families and providing new bounds and exact values.

Contribution

It introduces new bounds for the $(m,n)$-relative clique number across various graph families, including subcubic, bounded degree, planar, and triangle-free planar graphs, with exact values for subcubic graphs.

Findings

Exact $(m,n)$-relative clique number for subcubic graphs.

Improved bounds for bounded degree graphs.

Enhanced bounds for planar and triangle-free planar graphs.

Abstract

An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph 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 of an $(m, n)$-colored mixed graph $G$, introduced by Ne\v{s}et\v{r}il and Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of the smallest homomorphic image of $G$. Later Bensmail, Duffy and Sen [Graphs Combin. 2017] introduced another parameter related to the $(m, n)$-colored mixed chromatic number, namely, the $(m, n)$-relative clique number as the maximum cardinality of a vertex subset which, pairwise, must have distinct images with respect to any colored homomorphism. In this article, we study the $(m, n$)-relative clique number for the family of subcubic graphs, graphs with maximum degree $\Delta$, planar graphs and triangle-free planar graphs and provide new improved bounds in each of the cases. In particular, for subcubic graphs we provide exact value of the parameter.