29 research outputs found
Vertex decompositions of two-dimensional complexes and graphs
We investigate families of two-dimensional simplicial complexes defined in
terms of vertex decompositions. They include nonevasive complexes, strongly
collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary
and Palmer. We investigate the complexity of recognition problems for those
families and some of their combinatorial properties. Certain results follow
from analogous decomposition techniques for graphs. For example, we prove that
it is NP-complete to decide if a graph can be reduced to a discrete graph by a
sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug
Clique complexes and graph powers
We study the behaviour of clique complexes of graphs under the operation of
taking graph powers. As an example we compute the clique complexes of powers of
cycles, or, in other words, the independence complexes of circular complete
graphs.Comment: V3: final versio
Face numbers of down-sets
We compare various viewpoints on down-sets (simplicial complexes),
illustrating how the combinatorial inclusion-exclusion principle may serve as
an alternative to more advanced methods of studying their face numbers.Comment: 3 pages, accepted to Amer. Math. Monthly, v2: typos fixe
Transitivity is not a (big) restriction on homotopy types
For every simplicial complex K there exists a vertex-transitive simplicial
complex homotopy equivalent to a wedge of copies of K with some copies of the
circle. It follows that every simplicial complex can occur as a homotopy wedge
summand in some vertex-transitive complex. One can even demand that the
vertex-transitive complex is the clique complex of a Cayley graph or that it is
facet-transitive
Upper bound theorem for odd-dimensional flag triangulations of manifolds
We prove that among all flag triangulations of manifolds of odd dimension
2r-1 with sufficiently many vertices the unique maximizer of the entries of the
f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses
methods from extremal graph theory.Comment: Clarifications and new references, title has change