Transitive path decompositions of Cartesian products of complete graphs

TL;DR

This paper investigates highly symmetrical decompositions of the Cartesian product of complete graphs into paths, providing constructions that support longstanding conjectures in graph theory.

Contribution

It introduces a new construction for transitive path decompositions of $K_n \Box K_n$ when $n$ is an odd prime, advancing understanding of graph decompositions.

Findings

Supports Gallai's conjecture

Extends Ringel's conjecture

Constructs large paths in decompositions

Abstract

An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies of $H$) are preserved and transitively permuted by a group of automorphisms of $\Gamma$. This paper concerns transitive $H$-decompositions of the graph $K_n \Box K_n$ where $H$ is a path. When $n$ is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are considerably large compared to the number of vertices. Our main result supports well-known Gallai's conjecture and an extended version of Ringel's conjecture.