16 research outputs found

    Fractional clique decompositions of partite graphs

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    We give a minimum degree condition sufficient to ensure the existence of a fractionalKr-decomposition in a balancedr-partite graph (subject to some further simple necessary conditions). This generalizes the non-partite problem studied recently by Barber, Lo, Kühn, Osthus and the author, and the 3-partite fractionalK3-decomposition problem studied recently by Bowditch and Dukes. Combining our result with recent work by Barber, Kühn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional)Kr-decomposition in a balancedr-partite graph (subject to the same simple necessary conditions).</jats:p

    Resolution of the Oberwolfach problem

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    The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1K_{2n+1} into edge-disjoint copies of a given 22-factor. We show that this can be achieved for all large nn. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large nn.Comment: 28 page

    Threshold for Steiner triple systems

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    We prove that with high probability G(3)(n,n1+o(1))\mathbb{G}^{(3)}(n,n^{-1+o(1)}) contains a spanning Steiner triple system for n1,3(mod6)n\equiv 1,3\pmod{6}, establishing the tight exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.Comment: 22 pages, 1 figur

    The linear system for Sudoku and a fractional completion threshold

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    We study a system of linear equations associated with Sudoku latin squares. The coefficient matrix MM of the normal system has various symmetries arising from Sudoku. From this, we find the eigenvalues and eigenvectors of MM, and compute a generalized inverse. Then, using linear perturbation methods, we obtain a fractional completion guarantee for sufficiently large and sparse rectangular-box Sudoku puzzles

    Clique decompositions of multipartite graphs and completion of Latin squares

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    Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series

    Substructures in Latin squares

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    We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-nn Latin squares with no intercalate (i.e., no 2×22\times2 Latin subsquare) is at least (e9/4no(n))n2(e^{-9/4}n-o(n))^{n^{2}}. Equivalently, Pr[N=0]en2/4(n2)=e(1+o(1))EN\Pr\left[\mathbf{N}=0\right]\ge e^{-n^{2}/4- (n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}, where N\mathbf{N} is the number of intercalates in a uniformly random order-nn Latin square. In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant 0<δ10<\delta\le1 we have Pr[N(1δ)EN]=exp(Θ(n2))\Pr[\mathbf{N}\le(1-\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{2})) and for any constant δ>0\delta>0 we have Pr[N(1+δ)EN]=exp(Θ(n4/3(logn)2/3))\Pr[\mathbf{N}\ge(1+\delta)\mathbb{E}\mathbf{N}]=\exp(-\Theta(n^{4/3}(\log n)^{2/3})). Finally, we show that in almost all order-nn Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate 2×22\times2 subsquares with the same arrangement of symbols) is of order n4n^{4}, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring "how associative" the quasigroup associated with the Latin square is.Comment: 32 pages, 1 figur

    On the decomposition threshold of a given graph

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    We study the FF-decomposition threshold δF\delta_F for a given graph FF. Here an FF-decomposition of a graph GG is a collection of edge-disjoint copies of FF in GG which together cover every edge of GG. (Such an FF-decomposition can only exist if GG is FF-divisible, i.e. if e(F)e(G)e(F)\mid e(G) and each vertex degree of GG can be expressed as a linear combination of the vertex degrees of FF.) The FF-decomposition threshold δF\delta_F is the smallest value ensuring that an FF-divisible graph GG on nn vertices with δ(G)(δF+o(1))n\delta(G)\ge(\delta_F+o(1))n has an FF-decomposition. Our main results imply the following for a given graph FF, where δF\delta_F^\ast is the fractional version of δF\delta_F and χ:=χ(F)\chi:=\chi(F): (i) δFmax{δF,11/(χ+1)}\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}; (ii) if χ5\chi\ge 5, then δF{δF,11/χ,11/(χ+1)}\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}; (iii) we determine δF\delta_F if FF is bipartite. In particular, (i) implies that δKr=δKr\delta_{K_r}=\delta^\ast_{K_r}. Our proof involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory, Series
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