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arXiv · 2609.22928

An Odd Pfaffian Number

Priyanshu Pant·Ranveer Singh
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Abstract

The Pfaffian number of a graph is the minimum number of Pfaffians needed to obtain its perfect-matching polynomial by linear combination. In 2009, Norine conjectured that every Pfaffian number is a power of four. Miranda and Lucchesi disproved this conjecture in 2011 by constructing a graph of Pfaffian number six, and conjectured instead that every nontrivial Pfaffian number is even. We disprove their conjecture by proving that the Pfaffian number of $K_{3,3}\sqcup K_{3,3}$ is 13. We also construct a connected cubic bipartite matching-covered graph with Pfaffian number 13.

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Priyanshu Pant, Ranveer Singh. 2026-09-19. An Odd Pfaffian Number. https://arxiv.org/abs/2609.22928

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