[Paper Review] Graphs whose Eulerian trails have unique labels

Donggyu Kim, Rose McCarty+1 more|arXiv (Cornell University)|Mar 3, 2026

Advanced Graph Theory Research0 citations

TL;DR

The paper gives a structural and algorithmic characterization of when all Eulerian trails between two fixed vertices in a group-labeled undirected graph share the same label; in particular, 3-edge-connected parts are labeled over Z2^k, and there is a polynomial-time algorithm using a word-problem oracle.

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.

Motivation & Objective

• Understand when every Eulerian trail from a to b has the same label in a group-labeled graph.
• Provide a structural decomposition into 3-edge-connected parts labeled over Z2^k.
• Bridge the theory with an efficient algorithm using a word-problem oracle.
• Explore the limits of extending results beyond abelian groups and the Z2^k case.

Proposed method

• Develop a structure theorem: equivalence between identical-label Eulerian trails and 3-edge-connected components labeled over Z2^k after shifting.
• Prove the abelian case via a direct argument showing the generated group is Z2^k under appropriate shifting.
• Handle the 3-edge-connected case with a detailed inductive argument and edge-splitting techniques (Lemma 2.3, Proposition 3.1).
• Extend to general graphs by decomposing into cores (3-core) and reducing to core-valid instances (Theorem 4.3).
• Provide an algorithmic procedure to decide if all Eulerian circuits from a to b have the same label and to find differing labels when they exist (Theorem 1.2).

Experimental results

Research questions

• RQ1 When do all Eulerian trails from a to b have the same label in a group-labeled graph?
• RQ2 What structure do the 3-edge-connected components impose on the labeling?
• RQ3 Can the condition be reduced to labeling over a finite abelian group like Z2^k after shifts?
• RQ4 What is the complexity of deciding label-uniform Eulerian trails given a word-problem oracle?

Key findings

• A precise structural equivalence: all Eulerian trails from a to b have the same label if and only if, after shifting, each 3-edge-connected part is labeled by a group isomorphic to Z2^k.
• In the abelian case, this implies the whole graph is labeled over Z2^k.
• The main algorithm runs in time O(m^2)·φ(12m) to decide the property, where φ(n) is the word-problem oracle time, and can find two trails with different labels in O(m^4)·φ(12m).
• The results extend to processing graphs decomposed into cores, enabling practical handling of non-3-edge-connected graphs.
• If a feasible circuit with order >2 exists, there are two Eulerian trails from a to b with different labels.