On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Abstract

An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)-relative clique R of an (n, m)-graph G is a vertex subset of G for which no two distinct vertices of R get identified under any homomorphism of G. The (n, m)-relative clique number of G, denoted by ω r(n,m) (G), is the maximum |R| such that R is an (n, m)-relative clique of G. In practice, (n, m)relative cliques are often used for establishing lower bounds of (n, m)-chromatic number of graph families.Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that ω r(n,m) (G) ≤ 2(2n + m) 2 + 2 for any triangle-free planar (n, m)-graph G and that this bound is tight for all (n, m) = (0, 1). In this article, we positively settle this conjecture by improving the previous upper bound of ω r(n,m) (G) ≤ 14(2n+m) 2 +2 to ω r(n,m) (G) ≤ 2(2n+m) 2 +2, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of 2(2n + m) 2 + 2 for the (n, m)-chromatic number for the family of triangle-free planar graphs.