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.

翻译：$(n,m)$-图是指包含$n$种有向弧和$m$种无向边的图。两个$(n,m)$-图$G$与$H$之间的同态是一种保持邻接关系及其类型和方向的顶点映射。使得存在从$G$到$H$的同态的最小（按顶点数计）$H$的阶称为$G$的$(n,m)$-色数。此外，$(n,m)$-图$G$的$(n,m)$-相对团$R$是$G$的一个顶点子集，其中$R$中任意两个不同顶点在$G$的任何同态下都不会被等同。$G$的$(n,m)$-相对团数记为$\omega_{r(n,m)}(G)$，是满足$R$为$G$的$(n,m)$-相对团的最大$|R|$。实际上，$(n,m)$-相对团常用于建立图族$(n,m)$-色数的下界。推广Sopena [Discrete Mathematics 2016]在其定向着色最新综述中提出的一个未解决问题，Chakroborty、Das、Nandi、Roy与Sen [Discrete Applied Mathematics 2022]猜想：对任意无三角形平面$(n,m)$-图$G$，有$\omega_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$，且该界在$(n,m) \neq (0,1)$时是紧的。本文通过将$\omega_{r(n,m)}(G)$的上界从$14 (2n+m)^2 + 2$改进为$2 (2n+m)^2 + 2$，并构造达到此界的无三角形平面图实例，肯定地解决了该猜想。作为紧性证明的推论，我们还为无三角形平面图族建立了$(n,m)$-色数的新下界$2 (2n+m)^2 + 2$。