Defect Antichains and Multigraded Symbolic Defect Series of Edge Ideals under Graph Blow-ups

Abstract

In this paper, we study symbolic defect functions of edge ideals through finite antichains of exponent vectors. Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. For each symbolic degree $s$, we define the symbolic exponent region ${\mathcal{P}}_s(G)$, the ordinary exponent region ${\mathcal{O}}_s(G)$, and the symbolic defect antichain ${\mathcal{D}}_s(G)=\min ({\mathcal{P}}_s(G)\setminus {\mathcal{O}}_s(G))$, where the minimum is taken with respect to the componentwise partial order. We prove that ${\mathcal{D}}_s(G)$ gives a finite obstruction set controlling the minimal monomial generators of the quotient $I(G{)}^{(s)}/I(G{)}^s$. Our main result is a blow-up transfer formula. If $G^{\mathbf{n}}$ is the graph obtained from $G$ by replacing each vertex $v_i$ by an independent set of size $n_i$, then for every $s\ge 1$, \[ \operatorname{sdefect}(I(G^{\mathbf n}),s) = \sum_{\mathbf a\in \mathcal D_s(G)} \prod_{i=1}^{r} \binom{a_i+n_i-1}{n_i-1}. \] We further refine this formula to a multigraded symbolic defect series, which records the full multidegree distribution of the minimal generators of $I(G^{\mathbf{n}}{)}^{(s)}/I(G^{\mathbf{n}}{)}^s$. As applications, we classify the defect antichains of complete graphs in terms of integer partitions and derive explicit symbolic defect formulas for complete multipartite graphs, complete split graphs, and blow-ups of odd cycles. We also study symbolic defect antichains under graph joins and obtain polynomiality and rational generating-function consequences in the blow-up parameters. The results provide a unified antichain-based framework for symbolic defects of edge ideals and convert several previously case-by-case computations into consequences of a single transfer principle.

26 pages