Shellings, chordality, and Simon's conjecture

Abstract: A simplicial complex X is "shellable" if there exists an ordering of its facets that satisfies nice intersection properties. Shellability imposes strong topological and algebraic conditions on X and its Stanley-Reisner ring, and has been an important tool in geometric and algebraic combinatorics. Examples of shellable complexes include boundaries of simplicial polytopes and the independence complex of matroids. In general it is difficult (NP hard) to determine if a given complex is shellable, and X is said to be "extendably shellable" if a greedy algorithm always succeeds. A conjecture of Simon posits that the k-skeleton of a simplex on vertex set [n] is extendably shellable.

Simon's conjecture has been established for k=2 but until recently all other nontrivial cases were open. We show how the case k=n-3 follows from an application of chordal graphs and the notion of "exposed edges", and in fact prove that any shellable d-dimensional complex on at most d+3 vertices is extendably shellable. This leads to a notion of higher-dimensional chordality which connects Simon's conjecture to tools in commutative algebra and simple homotopy theory. We also explore other cases of Simon's conjecture and for instance prove that any vertex decomposable complex can be completed to a shelling of a simplex skeleton. Parts of this are joint work with Culertson, Guralnik, Stiller, and Oh.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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