A construction of smooth travel groupoids on finite graphs

TL;DR

This paper presents an algorithm to construct smooth travel groupoids on any finite graph, addressing a question about the existence of such structures on connected graphs.

Contribution

The paper introduces a method to construct smooth travel groupoids for all finite graphs, providing an answer to an open question in the field.

Findings

Algorithm successfully constructs smooth travel groupoids for finite graphs.

Addresses and resolves an open question about the existence of smooth travel groupoids on connected graphs.

Demonstrates the applicability of the construction to all finite graphs.

Abstract

A travel groupoid is an algebraic system related with graphs. In this paper, we give an algorithm to construct smooth travel groupoids for any finite graph. This algorithm gives an answer of L.~Nebesk$\acute{\mbox{y}}$'s question, "Does there exists a connected graph $G$ such that $G$ has no smooth travel groupoid?", in finite cases.