Pure vertex decomposable simplicial complex associated to graphs whose 5-cycles are chorded

Abstract

If a simplicial complex \(\Delta \) is vertex decomposable, then the n-sphere moment angle complex \(\mathcal {Z}_{\Delta ^{\vee }}(D^{n},S^{n-1})\) has the homotopy type of the wedges of spheres, where \(\Delta ^{\vee }\) is the Alexander dual of \(\Delta \). Furthermore, if \(\Delta \) is pure vertex decomposable, then its Stanley–Reisner ring \(k[\Delta ]\) is Cohen–Macaulay. Consequently, the vertex decomposable property is an interesting property from combinatorial, algebraic and topological point of view. In this paper, we characterize the pure vertex decomposable simplicial complexes associated to graphs whose 5-cycles have at least 4 chords.