2024/09/09 by Das, Kanoy Kumar, Amit Roy, Roy, Amit +2 · 1 citation
Computer Science · Mathematics · #05C70 #05E40 #13D02 #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Primary: 13C15 #Secondary: 13H10
paper · pdf · doi:10.48550/arxiv.2409.06021
openalex publication_date 2024/09/09 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28
Let I(G)[k] denote the kth square-free power of the edge ideal I(G) of a graph G. In this article, we provide a precise formula for the depth of I(G)[k] when G is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaulay forest G, the kth square-free power of I(G) is always Cohen-Macaulay, which is quite surprising since all ordinary powers of I(G) can never be Cohen-Macaulay unless G is a disjoint union of edges. Next, we give an exact formula for the regularity and tight bounds on the depth of square-free powers of edge ideals of cycles. In the case of whiskered cycles, we obtain tight bounds on the regularity and depth of square-free powers, which aids in identifying when such ideals have linear resolutions. Additionally, we compute depth of I(G)[2] when G is a cycle or whiskered cycle, and regularity of I(G)[2] when G is a whiskered cycle.