Density of linearity index in the interval of matching numbers

TL;DR

This work investigates the linearity properties of squarefree powers $I(G)^{[k]}$ of edge ideals and the associated linearity index $c(G)$, which satisfies $\nu_1(G)\le c(G)\le \nu(G)$. The authors construct, for every triple with $2 \le p \le c \le q$, a graph $G_{p,c,q}$ with $\nu_1(G_{p,c,q})=p$, $\nu(G_{p,c,q})=q$, and $c(G_{p,c,q})=c$ such that $I(G_{p,c,q})^{[k]}$ has linear quotients for all $k$ in $[c, q]$ while $I(G_{p,c,q})^{[k]}$ is not linearly related for any $k$ in $[1, c-1]$. The construction relies on adding whiskers to a joined graph $H_1*H_2$, with $H_2$ complete, yielding explicit graphs realizing all admissible triples. The results sharpen understanding of how combinatorial graph structure controls the homological behavior of edge ideals and connect to known cases (e.g., forests) where higher squarefree powers exhibit linear resolutions. Overall, the paper provides a complete realization of the prescribed three-parameter family and advances the combinatorial-algebraic theory of squarefree powers.

Abstract

Given integers $2 \leq p \leq c \leq q$, we construct a finite simple graph $G$ with $ν_1(G) = p$ and $ν(G) = q$ for which the squarefree power $I(G)^{[k]}$ of the edge ideal $I(G)$ of $G$ has linear quotients for each $c \leq k \leq q$ and is not linearly related for each $1 \leq k < c$, where $ν_1(G)$ is the induced matching number of $G$ and $ν(G)$ is the matching number of $G$.

Theorems & Definitions (11)

• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Definition 2.5
• Theorem 2.6
• proof