2025/06/02 by Amit Roy, Roy, Amit, Kamalesh Saha +1 · 2 citations
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Topological and Geometric Data Analysis #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2506.01810
For a non-negative integer k, let HSk(J(G)) denote the kth homological shift ideal of the vertex cover ideal J(G) of a graph G. For each k≥ 2, we construct a Cohen-Macaulay very well-covered graph Gk which is both Cohen-Macaulay bipartite and a whiskered graph so that HSk(J(G)) does not have a linear resolution. This contradicts several results as well as disproves a conjecture in [J. Algebra, 629, (2023), 76-108] and [Mediterr. J. Math., 21, 135 (2024)]. The graphs Gk are also examples of clique-whiskered graphs introduced by Cook and Nagel, which include Cohen-Macaulay chordal graphs, Cohen-Macaulay Cameron-Walker graphs, and clique corona graphs. Surprisingly, for Cohen-Macaulay chordal graphs, we can use a special ordering on the minimal generators to show that HSk(J(G)) has linear quotients for all k. Moreover, for all Cohen-Macaulay Cameron-Walker graphs and certain clique corona graphs, we show that HSk(J(G)) is weakly polymatroidal, and thus, has linear quotients for all k.