Betti Numbers of Cut Complexes of Squared Paths and a Recurrence Conjecture

TL;DR

This paper proves a recurrence relation for Betti numbers of cut complexes of squared paths, providing explicit formulas and polynomial properties, advancing understanding of their topological and combinatorial structure.

Contribution

It establishes an exact formula for Betti numbers of squared path cut complexes and confirms related conjectures, revealing polynomial patterns in their diagonal sequences.

Findings

Derived an explicit formula for Betti numbers of squared path cut complexes.

Confirmed conjectural closed forms for specific cases k=4,5.

Showed diagonal sequences are polynomials of degree r-1 with r-th finite difference zero.

Abstract

For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the disconnected $k$-vertex induced subgraphs of $G$. Bayer, Denker, Jeli\'c Milutinovi\'c, Sundaram, and Xue proved that the $k$-cut complex of the squared path $P_n^2$ is shellable for $n\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along every diagonal $n-k=r$. We prove the recurrence by giving the exact formula $\beta(k,n)=\binom{n-1}{k-1}-\sum_{j=0}^{\min\{k-1,n-k\}}\binom{k-1}{j}(n-k-j+1)+(n-k)$ for $r=n-k\ge3$. Equivalently, for fixed $r\ge3$, the diagonal sequence $B_r(k)=\beta(k,k+r)$ is a polynomial in $k$ of degree $r-1$, and therefore $\nabla^rB_r(k)=0$. The proof uses a complementary-face enumeration: among complements with size at least $k$, all bad complements have size $k$ or $k+1$, and they are, respectively, connected $k$-subsets of $P_n^2$ and intervals of length $k+1$. The same formula also proves the conjectural closed forms for $k=4,5$.