Square Property, Equitable Partitions, and Product-like Graphs

TL;DR

This paper explores how certain edge equivalence relations in graphs induce equitable vertex partitions, leading to quotient graphs with complex product structures, even when the original graph is prime.

Contribution

It establishes a connection between edge relations satisfying square conditions and equitable partitions, revealing new product-like structures in quotient graphs.

Findings

Edge relations induce equitable partitions

Quotient graphs can have rich product structures

Prime graphs can produce non-trivial quotient products

Abstract

Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of $G$, which in turn give rise to quotient graphs that can have a rich product structure even if $G$ itself is prime.