Transitive decompositions of graphs and their links with geometry and origami

TL;DR

This paper explores transitive decompositions of graphs, linking them to geometry and origami, and introduces a construction method using graph quotients with applications in modular origami.

Contribution

It provides a new method for constructing transitive graph decompositions via graph quotients and connects these concepts to geometric structures and origami.

Findings

A simple construction method for transitive decompositions using graph quotients

Connection established between transitive decompositions and partial linear spaces

Application demonstrated in modular origami design

Abstract

A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. In this paper we give some background to the study of transitive decompositions and highlight a connection with partial linear spaces. We then describe a simple method for constructing transitive decompositions using graph quotients, and we show how this may be used in an application to modular origami.