Prescribed and simultaneous failures of log-concavity in trees with perfect matchings

Abstract

We construct trees with perfect matchings and independence sequences that fail log-concavity near the independence number $α$. For every integer $k\geq1$, one family fails at $α-k$ and has a witness of order $O(k^2\log(k+2))$. For every integer $t\geq2$, a single tree has simultaneous failures at $α-1,α-3,\ldots,α-(2t-1)$, with a witness of order $O(t^3)$. All these trees satisfy $α=N/2$, where $N$ is the number of vertices, and contradict Conjecture 4.2 of Ramos and Sun. A sufficient condition uses a fixed tree and the numbers of attached leaves. For fixed $k$ or $t$, explicit bounds give geometric decay of $a_j^2/(a_{j-1}a_{j+1})$ at these indices as the trees grow; $a_j$ counts independent $j$-sets. The proofs count independent sets by empty matching pairs. The Lean proof assistant checks formal proofs of the parameterized results and their quantitative bounds; exact 90- and 180-vertex examples illustrate the constructions.