A method based on t-admissible subgraphs determines the regularity of the t-th symbolic power of cover ideals of graphs, applied to bipartite unicyclic graphs.
Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs
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abstract
Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \[ \depth\big(S/J(C_n)^{(t)}\big) = n - 1 - \left\lfloor \frac{tn}{2t+1} \right\rfloor \] for all $t \ge 2$ and $n \ge 3$, where $S = K[x_1,\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.
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math.AC 1years
2026 1verdicts
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Critical subgraphs and the regularity of symbolic powers of cover ideals of graphs
A method based on t-admissible subgraphs determines the regularity of the t-th symbolic power of cover ideals of graphs, applied to bipartite unicyclic graphs.