T-partition systems and travel groupoids on a graph

TL;DR

This paper introduces T-partition systems to characterize travel groupoids on graphs, providing a new framework to understand the relationship between algebraic structures and graph properties.

Contribution

It presents a novel characterization of travel groupoids on graphs using T-partition systems, extending Nebeský's finite graph results.

Findings

T-partition systems characterize travel groupoids on graphs.

A new algebraic-graph theoretical framework is established.

Provides conditions for the existence of travel groupoids on graphs.

Abstract

The notion of travel groupoids was introduced by L. Nebesk\'y in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph $G$ has a travel groupoid if the graph associated with the travel groupoid is equal to $G$. Nebesk\'y gave a characterization for finite graphs to have a travel groupoid. In this paper, we introduce the notion of T-partition systems on a graph and give a characterization of travel groupoids on a graph in terms of T-partition systems.