2,887 research outputs found
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
Among graphs, groups, and latin squares
A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex
Absorbing Set Analysis and Design of LDPC Codes from Transversal Designs over the AWGN Channel
In this paper we construct low-density parity-check (LDPC) codes from
transversal designs with low error-floors over the additive white Gaussian
noise (AWGN) channel. The constructed codes are based on transversal designs
that arise from sets of mutually orthogonal Latin squares (MOLS) with cyclic
structure. For lowering the error-floors, our approach is twofold: First, we
give an exhaustive classification of so-called absorbing sets that may occur in
the factor graphs of the given codes. These purely combinatorial substructures
are known to be the main cause of decoding errors in the error-floor region
over the AWGN channel by decoding with the standard sum-product algorithm
(SPA). Second, based on this classification, we exploit the specific structure
of the presented codes to eliminate the most harmful absorbing sets and derive
powerful constraints for the proper choice of code parameters in order to
obtain codes with an optimized error-floor performance.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1306.511
The stable roommates problem with globally-ranked pairs
We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to [Irving et al. 06] to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs
Bounded colorings of multipartite graphs and hypergraphs
Let be an edge-coloring of the complete -vertex graph . The
problem of finding properly colored and rainbow Hamilton cycles in was
initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied
since then. Recently it was extended to the hypergraph setting by Dudek, Frieze
and Ruci\'nski. We generalize these results, giving sufficient local (resp.
global) restrictions on the colorings which guarantee a properly colored (resp.
rainbow) copy of a given hypergraph .
We also study multipartite analogues of these questions. We give (up to a
constant factor) optimal sufficient conditions for a coloring of the
complete balanced -partite graph to contain a properly colored or rainbow
copy of a given graph with maximum degree . Our bounds exhibit a
surprising transition in the rate of growth, showing that the problem is
fundamentally different in the regimes and Our
main tool is the framework of Lu and Sz\'ekely for the space of random
bijections, which we extend to product spaces
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