Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals

Abstract

For a bipartite graph $G$ and an integer $t\ge 2$, let $\NM_t(G)$ be its $t$-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree $2t-3$. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that $\NM_t(G)$ has Leray number $2t-2$ whenever $G$ contains a cycle of length at least $2t$. Under the same hypothesis, Hochster's formula yields regularity $2t-1$ for the Stanley-Reisner ideal $I_{\NM_t(G)}$, together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If $G$ contains a $2t$-cycle, we also determine the maximal shifts of $I_{\NM_t(G)}$ through homological degree $|E(G)|-2t+1$. For $G=K_{r,s}$ with $2\le t\le r\le s$, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of $I_{\NM_t(K_{r,s})}$, thereby settling a conjecture on facet ideals of chessboard complexes.

13 pages