Optimal Enumeration of Eulerian Trails in Directed Graphs

2026-03-13

AI Summary

This study addresses the problem of efficiently enumerating all Eulerian trails in directed graphs. The authors propose a novel direct enumeration algorithm that achieves the optimal time complexity of $O(m + z_T)$, where $m$ is the number of edges and $z_T$ is the total number of Eulerian trails, without relying on the classical BEST theorem. This approach unconditionally improves upon the $O(m \cdot z_T)$ algorithm by Conte et al. and significantly outperforms BEST-based methods when $z_T = o(n^2)$. The algorithm applies to both simple directed graphs and directed multigraphs, and it is amenable to practical applications in domains such as bioinformatics and data privacy.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The BEST theorem, due to de Bruijn, van Aardenne-Ehrenfest, Smith, and Tutte, is a classical tool from graph theory that links the Eulerian trails in a directed graph $G=(V,E)$ with the arborescences in $G$. In particular, one can use the BEST theorem to count the Eulerian trails in $G$ in polynomial time. For enumerating the Eulerian trails in $G$, one could naturally resort to first enumerating the arborescences in $G$ and then exploiting the insight of the BEST theorem to enumerate the Eulerian trails in $G$: every arborescence in $G$ corresponds to at least one Eulerian trail in $G$. Instead, we take a simple and direct approach. Our central contribution is a remarkably simple algorithm to directly enumerate the $z_T$ Eulerian trails in $G$ in the \emph{optimal} $O(m + z_T)$ time. As a consequence, our result improves on an implementation of the BEST theorem for counting Eulerian trails in $G$ when $z_T=o(n^2)$, and, in addition, it unconditionally improves the combinatorial $O(m\cdot z_T)$-time algorithm of Conte et al. [FCT 2021] for the same task. Moreover, we show that, with some care, our algorithm can be extended to enumerate Eulerian trails in directed multigraphs in optimal time, enabling applications in bioinformatics and data privacy.

Problem

• Eulerian trails
• directed graphs
• optimal enumeration
• graph theory
• BEST theorem

Innovation

• Eulerian trails
• optimal enumeration
• directed graphs
• BEST theorem
• linear-time algorithm