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Eulerian Orientations and Hadamard Codes: A Novel Connection via Counting

Authors Shuai Shao , Zhuxiao Tang

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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Eulerian orientations
  • Hadamard codes
  • Counting problems
  • Tractable classes

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Abstract

We discover a novel connection between two classical mathematical notions, Eulerian orientations and Hadamard codes by studying the counting problem of Eulerian orientations (#EO) with local constraint functions imposed on vertices. We present two special classes of constraint functions and a chain reaction algorithm, and show that the #EO problem defined by each class alone is polynomial-time solvable by the algorithm. These tractable classes of functions are defined inductively, and quite remarkably the base level of these classes is characterized perfectly by the well-known Hadamard code. Thus, we establish a novel connection between counting Eulerian orientations and coding theory. We also prove a #P-hardness result for the #EO problem when constraint functions from the two tractable classes appear together.

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Shuai Shao and Zhuxiao Tang. Eulerian Orientations and Hadamard Codes: A Novel Connection via Counting. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 86:1-86:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.86

Author Details

Shuai Shao
  • School of Computer Science and Technology & Hefei National Laboratory, University of Science and Technology of China, Hefei, China
Zhuxiao Tang
  • School of the Gifted Young, University of Science and Technology of China, Hefei, China

Funding

Supported by the Innovation Program for Quantum Science and Technology, 2021ZD0302901.

Acknowledgements

The first author wants to thank Jin-Yi Cai and Zhiguo Fu for many valuable discussions. The authors want to thank anonymous reviewers for their helpful comments.

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