The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preserving vertex mappings of graphs with $n$ types of arcs and $m$ types of edges was initiated by Ne\v{s}et\v{r}il and Raspaud [Journal of Combinatorial Theory, Series B 2000]. Later, some attempts were made to generalize the switch operation that is popularly used in the study of signed graphs, and study its effect on the above mentioned homomorphism. In this article, we too provide a generalization of the switch operation on $(n,m)$-graphs, which to the best of our knowledge, encapsulates all the previously known generalizations as special cases. We approach to study the homomorphism with respect to the switch operation axiomatically. We prove some fundamental results that are essential tools in the further study of this topic. In the process of proving the fundamental results, we have provided yet another solution to an open problem posed by Klostermeyer and MacGillivray [Discrete Mathematics 2004]. We also prove the existence of a categorical product for $(n,m)$-graphs on with respect to a particular class of generalized switch which implicitly uses category theory. This is a counter intuitive solution as the number of vertices in the Categorical product of two $(n,m)$-graphs on $p$ and $q$ vertices has a multiple of $pq$ many vertices, where the multiple depends on the switch. This solves an open question asked by Brewster in the PEPS 2012 workshop as a corollary. We also provide a way to calculate the product explicitly, and prove general properties of the product. We define the analog of chromatic number for $(n,m)$-graphs with respect to generalized switch and explore the interrelations between chromatic numbers with respect to different switch operations. We find the value of this chromatic number for the family of forests using group theoretic notions.

翻译：$(n,m)$-图（即具有$n$类有向弧和$m$类无向边且满足邻接保持顶点映射的图）的同态研究由Nešetřil和Raspaud发起 [Journal of Combinatorial Theory, Series B 2000]。此后，研究者尝试推广在符号图研究中广泛使用的切换操作，并探讨其对上述同态的影响。本文同样对$(n,m)$-图提出一种切换操作的推广形式——据我们所知，该推广涵盖了所有已知的推广特例。我们采用公理化方法研究关于切换操作的同态性质，并证明了若干对后续研究具有基础性意义的核心结论。在证明基础结论的过程中，我们为Klostermeyer和MacGillivray提出的开放问题 [Discrete Mathematics 2004] 提供了另一种解答。此外，我们证明了在特定类广义切换下$(n,m)$-图存在范畴积（该证明隐含使用了范畴论方法）。这一结果具有反直觉性：两个顶点数分别为$p$和$q$的$(n,m)$-图的范畴积的顶点数为$pq$的整数倍，且倍数取决于切换操作的选择。作为推论，该结果解决了Brewster在PEPS 2012研讨会提出的开放问题。我们还给出了该积的显式计算方法，并证明了积的一般性质。我们定义了$(n,m)$-图关于广义切换的色数类比概念，并探讨了不同切换操作下色数之间的相互关系。最后，利用群论方法给出了森林族在该色数意义下的具体值。