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

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 $\mathcal{D}_s(G)=\min\big(\mathcal{P}_s(G)\setminus \mathcal{O}_s(G)\big)$, 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 $s\geq 1$, \[ \operatorname{sdefect}(I(G^{\mathbf n}),s) = \sum_{\mathbf a\in \mathcal D_s(G)} \prod_{i=1}^{r} \binom{a_i+n_i-1}{n_i-1}. \] 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.