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arXiv · 2509.19683

The vertex covers, Betti numbers and projective dimensions of perfect binary trees

Nguyen Thu Hang·Tran Duc Dung·Do Van Kien
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Abstract

Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of $S/I$ and show that the total Betti number of $S/I$ at the highest homological degree always equals one.

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Nguyen Thu Hang, Tran Duc Dung, Do Van Kien. 2025-09-24. The vertex covers, Betti numbers and projective dimensions of perfect binary trees. https://arxiv.org/abs/2509.19683

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