6 research outputs found
Non-ridge-chordal complexes whose clique complex has shellable Alexander dual
A recent conjecture that appeared in three papers by Bigdeli--Faridi,
Dochtermann, and Nikseresht, is that every simplicial complex whose clique
complex has shellable Alexander dual, is ridge-chordal. This strengthens the
long-standing Simon's conjecture that the -skeleton of the simplex is
extendably shellable, for any . We show that the stronger conjecture has a
negative answer, by exhibiting an infinite family of counterexamples.Comment: Substantial improvements. To appear on Journal of Combinatorial
Theory, Series
Extremal Examples of Collapsible Complexes and Random Discrete Morse Theory
We present extremal constructions connected with the property of simplicial
collapsibility.
(1) For each , there are collapsible (and shellable) simplicial
-complexes with only one free face. Also, there are non-evasive
-complexes with only two free faces. (Both results are optimal in all
dimensions.)
(2) Optimal discrete Morse vectors need not be unique. We explicitly
construct a contractible, but non-collapsible -dimensional simplicial
complex with face vector that admits two distinct
optimal discrete Morse vectors, and . Indeed, we show
that in every dimension there are contractible, non-collapsible
simplicial -complexes that have and
as distinct optimal discrete Morse vectors.
(3) We give a first explicit example of a (non-PL) -manifold, with face
vector , that is collapsible but
not homeomorphic to a ball.
Furthermore, we discuss possible improvements and drawbacks of random
approaches to collapsibility and discrete Morse theory. We will introduce
randomized versions \texttt{random-lex-first} and \texttt{random-lex-last} of
the \texttt{lex-first} and \texttt{lex-last} discrete Morse strategies of
\cite{BenedettiLutz2014}, respectively --- and we will see that in many
instances the \texttt{random-lex-last} strategy works significantly better than
Benedetti--Lutz's (uniform) \texttt{random} strategy.
On the theoretical side, we prove that after repeated barycentric
subdivisions, the discrete Morse vectors found by randomized algorithms have,
on average, an exponential (in the number of barycentric subdivisions) number
of critical cells asymptotically almost surely.Comment: 25 pages, 9 figures, 2 tables; revised Section