An Odd Pfaffian Number

@misc{pant2026pfaffian,
  title = {{An Odd Pfaffian Number}},
  author = {Priyanshu Pant and Ranveer Singh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22928},
  eprint = {2609.22928},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22928},
}

TY  - UNPB
TI  - An Odd Pfaffian Number
AU  - Priyanshu Pant
AU  - Ranveer Singh
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22928
UR  - https://arxiv.org/abs/2609.22928
AB  - 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_3,3sqcup K_3,3 is 13. We also construct a connected cubic bipartite matching-covered graph with Pfaffian number 13.
ER  -