论文标题
（n，m）图的同态相对于广义开关
Homomorphisms of (n,m)-graphs with respect to generalised switch
论文作者
论文摘要
$（n，m）$ - 图的同态性研究的研究，即保存图形的邻近映射，具有$ n $ n $类型的弧线和$ m $类型的边缘类型，由Nešet树和Raspaud在2000年发起。后来，进行了一些尝试，这些尝试是为了推动了对图像的开关操作，并进行了研究。 在本文中，我们也提供了$（n，m）$图的开关操作的概括，据我们所知，这将所有先前已知的概括都封装为特殊情况。我们对开关操作的同态研究进行了研究。我们证明了一些基本结果，这些结果是对该主题进一步研究的重要工具。在证明基本结果的过程中，我们为Klostermeyer和MacGillivray在2004年提出的一个开放问题提供了另一种解决方案。我们还证明，与特定类别的广义开关相对于隐含的类别理论的特定类别类别的类别，我们还证明了$（n，m）$图形的分类产品的存在。这是一个反直观的解决方案，作为两个$ p $和$ q $ dertices上两个$（n，m）$ - 图形的分类产品中的顶点的数量，其中有$ pq $的多个顶点，其中多个取决于交换机。这解决了Brewster在PEPS 2012研讨会上提出的一个开放问题，作为推论。我们还提供了一种明确计算产品的方法，并证明了产品的一般特性。我们将$（n，m）$的色度类似物定义为广义开关，并探索相对于不同开关操作的色数之间的相互关系。我们使用群体理论观念发现了该色数对森林家族的价值。
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.
