Graphs whose Eulerian trails have unique labels

TL;DR

This paper characterizes the structure of undirected labeled graphs where all Eulerian trails between two vertices have identical labels, revealing a connection to groups isomorphic to powers of Z2 and providing a polynomial-time algorithmic solution.

Contribution

It provides a precise structural characterization of graphs with unique-labeled Eulerian trails and reduces the problem to the group word problem, advancing understanding in graph labeling and group theory.

Findings

Each 3-connected component is labeled over a group isomorphic to Z2^k.

The problem reduces to the group word problem in polynomial time.

Structural conditions determine when all Eulerian trails have the same label.

Abstract

Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.