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 \(\omega _{r(n,m)}(G)\) , 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 (Discr Math 339(7):1993–2005, 2016) in his latest survey on oriented coloring, Chakroborty et al. (Discr Appl Math 324:29–40, 2023) conjectured that \(\omega _{r(n,m)}(G) \le 2 (2n+m)^2 + 2\) for any triangle-free planar (n, m)-graph G and that this bound is tight for all \((n,m) \ne (0,1)\) . In this article, we positively settle this conjecture by improving the previous upper bound of \(\omega _{r(n,m)}(G) \le 14 (2n+m)^2 + 2\) to \(\omega _{r(n,m)}(G) \le 2 (2n+m)^2 + 2\) , 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 of the family of triangle-free planar graphs for \(2n+m \ge 3.\)
