Shellability of Polyhedral Joins of Simplicial Complexes and Its Application to Graph Theory

DOI10.37236/11295arXiv2205.03869MaRDI QIDQ6398542

Kengo Okura

Publication date: 8 May 2022

Abstract: We investigate the shellability of the polyhedral join $mathcal{Z}_M^{*}(K,L)$ of simplicial complexes $K,M$ and a subcomplex $LsubsetK$. We give sufficient conditions and necessary conditions on $(K,L)$ for $mathcal{Z}_M^{*}(K,L)$ being shellable. In particular, we show that for some pairs $(K,L)$, $mathcal{Z}_M^{*}(K,L)$ becomes shellable regardless of whether $M$ is shellable or not. Polyhedral joins can be applied to graph theory as the independence complex of a certain generalized version of lexicographic products of graphs which we define in this paper. The graph obtained from two graphs $G,H$ by attaching one copy of $H$ to each vertex of $G$ is a special case of this generalized lexicographic product and we give a result on the shellability of the independence complex of this graph by applying the above results.

Mathematics Subject Classification ID

Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes (13F55) Topological properties in algebraic geometry (14F45) Combinatorial aspects of simplicial complexes (05E45) Graph operations (line graphs, products, etc.) (05C76)

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