On coloring parameters of triangle-free planar $(n,m)$-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 $\omega_{r(n,m)}(G)$, 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 [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that $\omega_{r(n,m)}(G) \leq 2 (2n+m)² + 2$ for any triangle-free planar $(n,m)$-graph $G$ and that this bound is tight for all $(n,m) \neq (0,1)$.In this article, we positively settle this conjecture by improving the previous upper bound of $\omega_{r(n,m)}(G) \leq 14 (2n+m)² + 2$ to $\omega_{r(n,m)}(G) \leq 2 (2n+m)² + 2$, 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.