2,887 research outputs found

    Parity of transversals of Latin squares

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    We introduce a notion of parity for transversals, and use it to show that in Latin squares of order 2mod42 \bmod 4, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving E1,,EnE_1,\dots, E_n, where EiE_i is the number of diagonals of a given Latin square that contain exactly ii different symbols. Let A(ij)A(i\mid j) denote the matrix obtained by deleting row ii and column jj from a parent matrix AA. Define tijt_{ij} to be the number of transversals in L(ij)L(i\mid j), for some fixed Latin square LL. We show that tabtcdmod2t_{ab}\equiv t_{cd}\bmod2 for all a,b,c,da,b,c,d and LL. Also, if LL has odd order then the number of transversals of LL equals tabt_{ab} mod 2. We conjecture that tac+tbc+tad+tbd0mod4t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4 for all a,b,c,da,b,c,d. 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 kk-regular bipartite graph on 2n2n vertices is divisible by 44 when nn is odd and k0mod4k\equiv0\bmod 4. We also show that perA(ac)+perA(bc)+perA(ad)+perA(bd)0mod4{\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4 for all a,b,c,da,b,c,d, when AA is an integer matrix of odd order with all row and columns sums equal to k2mod4k\equiv2\bmod4

    Chapter on The history of latin squares

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    Among graphs, groups, and latin squares

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    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

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    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

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    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

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    Let cc be an edge-coloring of the complete nn-vertex graph KnK_n. The problem of finding properly colored and rainbow Hamilton cycles in cc 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 GG. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring cc of the complete balanced mm-partite graph to contain a properly colored or rainbow copy of a given graph GG with maximum degree Δ\Delta. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes Δm\Delta \gg m and Δm\Delta \ll m 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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