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Stanley–Reisner ideals of higher independence complexes of chordal graphs.

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Title: Stanley–Reisner ideals of higher independence complexes of chordal graphs.
Authors: Das, Kanoy Kumar1 (AUTHOR) kanoydas0296@gmail.com, Roy, Amit1 (AUTHOR) amitiisermohali493@gmail.com, Saha, Kamalesh2 (AUTHOR) kamalesh.s@srmap.edu.in
Source: International Journal of Algebra & Computation. Jun2026, Vol. 36 Issue 4, p383-406. 24p.
Subjects: Graph theory, Commutative algebra, Cohen-Macaulay rings
Abstract: For t ≥ 2 , the t-independence complex Ind t (G) of a graph G is the collection of all A ⊆ V (G) such that each connected component of the induced subgraph G [ A ] has at most t − 1 vertices. The topology of Ind t (G) is intimately related to the combinatorial property of G. In this paper, we consider the Stanley–Reisner ideal J t (G) of Ind t (G) and focus on its algebraic properties. We prove that for a chordal graph G and for all t, reg (R / J t (G)) = (t − 1) ν t (G) and pd (R / J t (G)) = bight (J t (G)) , where ν t (G) denotes the induced matching number of the corresponding hypergraph of J t (G) , and reg , pd and bight stand for the regularity, projective dimension and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley–Reisner ideal of the t-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen–Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs. [ABSTRACT FROM AUTHOR]
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Abstract:For t ≥ 2 , the t-independence complex Ind t (G) of a graph G is the collection of all A ⊆ V (G) such that each connected component of the induced subgraph G [ A ] has at most t − 1 vertices. The topology of Ind t (G) is intimately related to the combinatorial property of G. In this paper, we consider the Stanley–Reisner ideal J t (G) of Ind t (G) and focus on its algebraic properties. We prove that for a chordal graph G and for all t, reg (R / J t (G)) = (t − 1) ν t (G) and pd (R / J t (G)) = bight (J t (G)) , where ν t (G) denotes the induced matching number of the corresponding hypergraph of J t (G) , and reg , pd and bight stand for the regularity, projective dimension and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley–Reisner ideal of the t-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen–Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs. [ABSTRACT FROM AUTHOR]
ISSN:02181967
DOI:10.1142/S021819672650013X