Odd-cycle defects in the Alon-Friedland bound

Mohsen Aliabadi; Elliot Krop

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arXiv · 2607.14134

Odd-cycle defects in the Alon-Friedland bound

Mohsen Aliabadi·Elliot Krop

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Abstract

Let $G$ be a finite simple graph. We study perfect matchings through two complementary viewpoints: reductions to bipartite permanent terms and the directed cycle covers counted by the ordinary adjacency permanent. The main identity is an explicit odd-cycle-indexed form of the classical cycle-cover expansion: it separates $\perfmat(G)^2$, the even-cycle-cover contribution coming from superposing two perfect matchings, from the contribution of cycle covers containing odd cycles. This gives a nonnegative odd-cycle defect $\delta_{\mathrm{odd}}(G)$. Combining the identity with the Bregman--Minc inequality yields a structural refinement of the Alon--Friedland degree-sequence bound. We prove product, positivity, and fractional-perfect-matching interpretations for the defect; show that for $K_{2n}$ the defect asymptotically accounts for almost the entire Bregman--Minc target; and derive from this a derangement identity. We also study near equality in the Alon--Friedland bound: we classify all one-edge perturbations of the extremal graphs, record the uniform obstruction coming from $K_4$, and formulate a sharp bounded-degree gap problem whose two natural candidate extremal graphs cross between maximum degrees $9$ and $10$.

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Mohsen Aliabadi, Elliot Krop. 2026-07-09. Odd-cycle defects in the Alon-Friedland bound. https://arxiv.org/abs/2607.14134

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