171 research outputs found
Generating strings for bipartite Steinhaus graphs
AbstractLet b(n) be the number of bipartite Steinhaus graphs with n vertices. We show that b(n) satisfies the recurrence, b(2) = 2, b(3) = 4, and for k ⩾ 2, b(2k + 1) = 2b(k + 1) + 1, b(2k) = b(k) + b(k + 1). Thus b(n) ⩽ 52n − 72 with equality when n is one more than a power of two. To prove this recurrence, we describe the possible generating strings for these bipartite graphs
Regular Steinhaus graphs of odd degree
A Steinhaus matrix is a binary square matrix of size which is symmetric,
with diagonal of zeros, and whose upper-triangular coefficients satisfy
for all . Steinhaus matrices
are determined by their first row. A Steinhaus graph is a simple graph whose
adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem,
due to Dymacek, which states that even Steinhaus graphs, i.e. those with all
vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek
conjectured that the complete graph on two vertices is the only regular
Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if
is a Steinhaus matrix associated with a regular
Steinhaus graph of odd degree then its sub-matrix is a multi-symmetric matrix, that is a doubly-symmetric matrix where each
row of its upper-triangular part is a symmetric sequence. We prove that the
multi-symmetric Steinhaus matrices of size whose Steinhaus graphs are
regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend
on parameters for all even numbers , and on
parameters in the odd case. This result permits us
to verify the Dymacek's conjecture up to 1500 vertices in the odd case.Comment: 16 page
Towards random uniform sampling of bipartite graphs with given degree sequence
In this paper we consider a simple Markov chain for bipartite graphs with
given degree sequence on vertices. We show that the mixing time of this
Markov chain is bounded above by a polynomial in in case of {\em
semi-regular} degree sequence. The novelty of our approach lays in the
construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain
explicitly our main contribution and corrected a serious flaw in the cycle
decompositio
Balanced simplices
An additive cellular automaton is a linear map on the set of infinite
multidimensional arrays of elements in a finite cyclic group
. In this paper, we consider simplices appearing in the
orbits generated from arithmetic arrays by additive cellular automata. We prove
that they are a source of balanced simplices, that are simplices containing all
the elements of with the same multiplicity. For any
additive cellular automaton of dimension or higher, the existence of
infinitely many balanced simplices of appearing in
such orbits is shown, and this, for an infinite number of values . The
special case of the Pascal cellular automata, the cellular automata generating
the Pascal simplices, that are a generalization of the Pascal triangle into
arbitrary dimension, is studied in detail.Comment: 33 pages ; 11 figures ; 1 tabl
Davies-trees in infinite combinatorics
This short note, prepared for the Logic Colloquium 2014, provides an
introduction to Davies-trees and presents new applications in infinite
combinatorics. In particular, we give new and simple proofs to the following
theorems of P. Komj\'ath: every -almost disjoint family of sets is
essentially disjoint for any ; is the union of
clouds if the continuum is at most for any ;
every uncountably chromatic graph contains -connected uncountably chromatic
subgraphs for every .Comment: 8 pages, prepared for the Logic Colloquium 201
Towards random uniform sampling of bipartite graphs with given degree sequence
In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on n vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in n in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method
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