On relative clique number of triangle-free planar (n,m)-graphs
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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.
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