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Lower Bounds for the Pfaffian Number of Graphs

Authors Enrique Junchaya , Alberto Alexandre Assis Miranda , Cláudio Leonardo Lucchesi

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Matchings and factors
Keywords
  • Perfect matchings
  • pfaffian graphs
  • k-pfaffian graphs
  • pfaffian number of graphs

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Abstract

The number of perfect matchings of a k-pfaffian graph can be counted by computing a linear combination of the pfaffians of k matrices. The pfaffian number of a graph G is the smallest integer k such that G is k-pfaffian. We present the first known lower bounds for the pfaffian number of graphs. As an intermediate step, we prove an upper bound for the rank of two matrices related to their Khatri-Rao product. One of the consequences of the found lower bounds is the existence of graphs whose pfaffian numbers are arbitrarily large.

Cite As Get BibTex

Enrique Junchaya, Alberto Alexandre Assis Miranda, and Cláudio Leonardo Lucchesi. Lower Bounds for the Pfaffian Number of Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.28

Author Details

Enrique Junchaya
  • Universidade de São Paulo, Brazil
Alberto Alexandre Assis Miranda
  • Instituto Federal do Norte de Minas Gerais, Montes Claros, Brazil
Cláudio Leonardo Lucchesi
  • Universidade de São Paulo, Brazil

Funding

  • Junchaya, Enrique: funded by São Paulo Research Foundation (FAPESP), Brazil. Process number #2024/02636-4.

Acknowledgements

We want to thank the referees for their useful comments.

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