Lower Bounds for the Pfaffian Number of Graphs
Combinatorics
2026-03-03 v1 Discrete Mathematics
Abstract
The number of perfect matchings of a -pfaffian graph can be counted by computing a linear combination of the pfaffians of matrices. The pfaffian number of a graph is the smallest integer such that is -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, a result of independent relevance. One of the consequences of these results is the existence of graphs whose pfaffian numbers are arbitrarily large.
Cite
@article{arxiv.2603.00334,
title = {Lower Bounds for the Pfaffian Number of Graphs},
author = {Enrique Junchaya and Alberto Alexandre Assis Miranda and Cláudio L. Lucchesi},
journal= {arXiv preprint arXiv:2603.00334},
year = {2026}
}
Comments
14 pages, 3 figures. Submitted