Homomorphisms of (n,m)-graphs with respect to generalised switch

TL;DR

This paper generalizes the switch operation for (n,m)-graphs, studies homomorphisms under this operation, solves open problems, and explores categorical products and chromatic numbers in this context.

Contribution

It introduces a comprehensive generalization of the switch operation on (n,m)-graphs, proves fundamental results, and explores categorical products and chromatic numbers.

Findings

Provided a solution to an open problem by Klostermeyer and MacGillivray.

Proved the existence of a categorical product for (n,m)-graphs with respect to a class of switches.

Calculated the chromatic number for forests using group theory.

Abstract

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 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.