(n,m)-图关于一般化切换的同态
Homomorphisms of (n,m)-graphs with respect to generalised switch
AI总结：
本文研究(n,m)-图关于一般化切换的同态，提出新的切换操作推广，并证明了范畴积的存在及色数的计算方法。
摘要：
(n,m)-图的同态研究，即保持邻接的顶点映射，最初由Nešetřil和Raspaud于2000年发起。后来，一些尝试将切换操作推广到研究符号图时广泛使用的切换操作，并研究其对上述同态的影响。在本文中，我们为(n,m)-图提供了切换操作的推广，据我们所知，这涵盖了所有之前已知的推广作为特殊情况。我们通过公理方法研究切换操作下的同态。我们证明了一些基本结果，这些结果是进一步研究该主题的重要工具。在证明基本结果的过程中，我们提供了一个新的解决方案，以解决Klostermeyer和MacGillivray于2004年提出的一个开放问题。我们还证明了针对特定类别的通用切换的(n,m)-图的范畴积存在，这隐含地使用了范畴论。这是一个反直觉的解决方案，因为两个具有p和q个顶点的(n,m)-图的范畴积的顶点数有pq倍的顶点，其中倍数取决于切换。这解决了Brewster在2012年PEPS研讨会中提出的一个开放问题作为推论。我们还提供了一种计算积的方法，并证明了积的一般性质。我们定义了针对通用切换的(n,m)-图的色数的类比，并探讨了不同切换操作下的色数之间的相互关系。我们利用群论概念确定了森林家族的色数值。
英文摘要：
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 Nešetřil and Raspaud in 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 the study of homomorphisms 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 in 2004. We also prove the existence of a categorical product for $(n,m)$-graphs 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.
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