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Some algebraic properties of \(t\)-clique ideals. (English) Zbl 1423.13115

Suppose that \(k\) is a field, \(R=k[x_1,\ldots,x_n]\) and \(G\) is a simple graph with vertex set \(\{x_1, \ldots, x_n\}\). A \(t\)-clique of \(G\) means a set of \(t\) vertices of \(G\) which are pairwise adjacent. Let \(K_t(G)\) be the ideal of \(R\) generated by the set \(\{x_{i_1}\cdots x_{i_t}|\{x_{i_1},\ldots,x_{i_t}\}\) is a \(t\)-clique of \(G\}\). Then \(K_t(G)\) is called the \(t\)-clique ideal of \(G\). Note that \(K_2(G)\) is the well-known edge ideal of \(G\). Also consider the corresponding simplicial complex \[\Delta_{K_t(G)}=\{F\subseteq \{x_1,\ldots, x_n\} \text{ such that }F\text{ does not contain any }t\text{-clique of }G\}. \] Note \(K_t(G)\) is the Stanley-Reisner ideal of \(\Delta_{K_t(G)}\).
In this paper, the authors first study when \(K_t(G)\) or its powers have linear quotients or a linear resolution. In particular, they show that if \(G\) is a path and \(I=K_{t}(G^c)\), where \(G^c\) denotes the complement of \(G\), then \(I^m\) has linear quotients and hence a \(tm\)-linear resolution, for each positive integer \(m\). The authors also present conditions under which \(\Delta_{K_t(G)}\) is shellable. Finally they prove that if \(G\) is a double broom graph (a graph obtained by attaching some pendant vertices to the end vertices of a path), then \(\Delta_{K_t(G)}\) is vertex decomposable (and hence shellable) for each positive integer \(t\).

MSC:

13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05E40 Combinatorial aspects of commutative algebra
05E45 Combinatorial aspects of simplicial complexes
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)

Software:

Macaulay2
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Full Text: DOI

References:

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