Irreducible Non-Metrizable Path Systems in Graphs

TL;DR

This paper constructs an infinite family of non-metrizable irreducible path systems within Paley graphs, advancing understanding of complex path structures in graph theory.

Contribution

It introduces a new class of irreducible path systems that are non-metrizable, specifically constructed on Paley graphs, expanding the theoretical landscape.

Findings

Constructed an infinite family of such path systems.

Demonstrated non-metrizability of these path systems.

Showed irreducibility in the context of Paley graphs.

Abstract

A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an infinite family of non-metrizable irreducible path systems defined on certain Paley graphs.