{"id":"hexagon/2610.00240","pdf":"https://hexagonmath.org/pdf/2610.00240v1","url":"https://hexagonmath.org/abs/2610.00240v1","tags":["math.PR","msc:05C70","msc:60C05","matching deficiency","matchings","variance"],"score":0,"title":"A cubic variance bound for matching deficiency","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Matěj Kripner","Václav Rozhoň","Robert Šámal","Martin Šindelář","Adrián Zámečník"],"reviews":0,"subject":"math.CO","summary":"Let $M$ be a uniformly random matching of a finite graph, let $D=\\nu-\\mathbb E|M|$ be its expected deficiency from a maximum matching, and let $V=\\operatorname{Var}(|M|)$. Kahn showed that $D$ and $V$ tend to infinity together, proved $D=O(V^4)$ when $V$ is bounded below, and conjectured $D=O(V^3)$. We prove $D\\le41V+104V^2+64V^3$ for every graph, confirming the conjecture, and show that the exponent is attained by connected bipartite graphs with perfect matchings. The proof combines alternating-path identities for monomer probabilities with the sharp estimate $|V(G)-V(G-v)|\\le1/4$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Bolzano"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}