Upper Embeddability of Graphs and Products of Transpositions Associated with Edges

TL;DR

This paper characterizes graphs where a specific edge ordering results in a full cyclic permutation of vertices, linking graph embeddings to permutation properties, and provides a counterexample related to Eden's question about identity permutations.

Contribution

It offers a new characterization of such graphs via embeddings and addresses Eden's question with a counterexample.

Findings

Characterization of graphs with a linear order producing a full cyclic permutation

Counterexample to Eden's question about identity permutations

Connection between graph embeddings and permutation properties

Abstract

Given a graph, we associate each edge with the transposition which exchanges the endvertices. Fixing a linear order on the edge set, we obtain a permutation of the vertices. D\'enes proved that the permutation is a full cyclic permutation for any linear order if and only if the graph is a tree. In this article, we characterize graphs having a linear order such that the associated permutation is a full cyclic permutation in terms of graph embeddings. Moreover, we give a counter example for Eden's question about an edge ordering whose associated permutation is the identity.