316 research outputs found
Spheres are rare
We prove that triangulations of homology spheres in any dimension grow much
slower than general triangulations. Our bound states in particular that the
number of triangulations of homology spheres in 3 dimensions grows at most like
the power 1/3 of the number of general triangulations.Comment: 14 pages, 1 figur
Connected Domination in Plane Triangulations
A set of vertices of a graph such that each vertex of is either in
the set or is adjacent to a vertex in the set is called a dominating set of
. If additionally, the set of vertices induces a connected subgraph of
then the set is a connected dominating set of . The domination number
of is the smallest number of vertices in a dominating set of
, and the connected domination number of is the smallest
number of vertices in a connected dominating set of . We find the connected
domination numbers for all triangulations of up to thirteen vertices. For , (mod 3), we find graphs of order and
. We also show that the difference
can be arbitrarily large.Comment: 12 pages, 10 figures, 1 tabl
Generating punctured surface triangulations with degree at least 4
As a sequel of a previous paper by the authors, we present here
a generating theorem for the family of triangulations of an arbitrary
punctured surface with vertex degree ≥ 4. The method is based on a
series of reversible operations termed reductions which lead to a minimal
set of triangulations in such a way that all intermediate triangulations
throughout the reduction process remain within the family. Besides contractible edges and octahedra, the reduction operations act on two new
configurations near the surface boundary named quasi-octahedra and
N-components. It is also observed that another configuration called
M-component remains unaltered under any sequence of reduction operations. We show that one gets rid of M-components by flipping appropriate edges
Random tensor models in the large N limit: Uncoloring the colored tensor models
Tensor models generalize random matrix models in yielding a theory of
dynamical triangulations in arbitrary dimensions. Colored tensor models have
been shown to admit a 1/N expansion and a continuum limit accessible
analytically. In this paper we prove that these results extend to the most
general tensor model for a single generic, i.e. non-symmetric, complex tensor.
Colors appear in this setting as a canonical book-keeping device and not as a
fundamental feature. In the large N limit, we exhibit a set of Virasoro
constraints satisfied by the free energy and an infinite family of
multicritical behaviors with entropy exponents \gamma_m=1-1/m.Comment: 15 page
Twin-width of graphs on surfaces
Twin-width is a width parameter introduced by Bonnet, Kim, Thomass\'e and
Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic
applications. We prove that the twin-width of every graph embeddable in a
surface of Euler genus is , which is asymptotically best
possible as it asymptotically differs from the lower bound by a constant
multiplicative factor. Our proof also yields a quadratic time algorithm to find
a corresponding contraction sequence. To prove the upper bound on twin-width of
graphs embeddable in surfaces, we provide a stronger version of the Product
Structure Theorem for graphs of Euler genus that asserts that every such
graph is a subgraph of the strong product of a path and a graph with a
tree-decomposition with all bags of size at most eight with a single
exceptional bag of size
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