180 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A proof of the Ryser-Brualdi-Stein conjecture for large even nn

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    A Latin square of order nn is an nn by nn grid filled using nn symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order nn contains a transversal with n1n-1 cells, and a transversal with nn cells if nn is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order nn has a transversal with nO(logn/loglogn)n-O(\log n/\log\log n) cells. Here, we show, for sufficiently large nn, that every Latin square of order nn has a transversal with n1n-1 cells. We also apply our methods to show that, for sufficiently large nn, every Steiner triple system of order nn has a matching containing at least (n4)/3(n-4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3O(logn/loglogn)n/3-O(\log n/\log\log n) edges, and proves a conjecture of Brouwer from 1981 for large nn.Comment: 71 pages, 13 figure

    Counting spanning subgraphs in dense hypergraphs

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    We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each k2k\geq 2 and 1k11\leq \ell\leq k-1, we show that every kk-graph on nn vertices with minimum codegree at least \cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,} contains exp(nlognΘ(n))\exp(n\log n-\Theta(n)) Hamilton \ell-cycles as long as (k)n(k-\ell)\mid n. When (k)k(k-\ell)\mid k this gives a simple proof of a result of Glock, Gould, Joos, K\"uhn and Osthus, while, when (k)k(k-\ell)\nmid k this gives a weaker count than that given by Ferber, Hardiman and Mond or, when <k/2\ell<k/2, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Paths and cycles in graphs and hypergraphs

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    In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles. A kk-uniform tight cycle Cn(k)C^{(k)}_n is a kk-uniform hypergraph on nn vertices with a cyclic ordering of its vertices such that the edges are all kk-sets of consecutive vertices in the ordering. We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to kk-uniform hypergraphs and prove results in the 4- and 5-uniform case. For a kk-uniform hypergraph~HH, the Ramsey number r(H){r(H)} is the smallest integer NN such that any 2-edge-colouring of the complete kk-uniform hypergraph on NN vertices contains a monochromatic copy of HH. We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that r(Cn(4))r(C^{(4)}_n) = (5+oo(1))nn. We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any γ\gamma >0 and kk \geq 3 asymptotically almost surely, every subgraph of the binomial random kk-uniform hypergraph G(k)(n,nγ1)G^{(k)}(n, n^{\gamma -1}) in which all (k1)(k-1)-sets are contained in at least (12+2γ)pn(\frac{1}{2}+2\gamma)pn edges has a tight Hamilton cycle. A random graph model on a host graph HH is said to be 1-independent if for every pair of vertex-disjoint subsets A,BA,B of E(H)E(H), the state of edges (absent or present) in AA is independent of the state of edges in BB. We show that pp = 4 - 23\sqrt{3} is the critical probability such that every 1-independent graph model on Z2×Kn\mathbb{Z}^2 \times K_n where each edge is present with probability at least pp contains an infinite path

    Universality for transversal Hamilton cycles

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    Let G={G1,,Gm}\mathbf{G}=\{G_1, \ldots, G_m\} be a graph collection on a common vertex set VV of size nn such that δ(Gi)(1+o(1))n/2\delta(G_i) \geq (1+o(1))n/2 for every i[m]i \in [m]. We show that G\mathbf{G} contains every Hamilton cycle pattern. That is, for every map χ:[n][m]\chi: [n] \to [m] there is a Hamilton cycle whose ii-th edge lies in Gχ(i)G_{\chi(i)}.Comment: 18 page

    Counting oriented trees in digraphs with large minimum semidegree

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    Let TT be an oriented tree on nn vertices with maximum degree at most eo(logn)e^{o(\sqrt{\log n})}. If GG is a digraph on nn vertices with minimum semidegree δ0(G)(12+o(1))n\delta^0(G)\geq(\frac12+o(1))n, then GG contains TT as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree o(n/logn)o(n/\log n)). This generalizes the corresponding result by Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. We investigate the natural question how many copies of TT the digraph GG contains. Our main result states that every such GG contains at least Aut(T)1(12o(1))nn!|Aut(T)|^{-1}(\frac12-o(1))^nn! copies of TT, which is optimal. This implies the analogous result in the undirected case.Comment: 24 page

    Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

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    We prove that for nNn \in \mathbb N and an absolute constant CC, if pClog2n/np \geq C\log^2 n / n and Li,j[n]L_{i,j} \subseteq [n] is a random subset of [n][n] where each k[n]k\in [n] is included in Li,jL_{i,j} independently with probability pp for each i,j[n]i, j\in [n], then asymptotically almost surely there is an order-nn Latin square in which the entry in the iith row and jjth column lies in Li,jL_{i,j}. The problem of determining the threshold probability for the existence of an order-nn Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides an upper bound which is tight up to a factor of logn\log n and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and 11-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a 11-factorization of a nearly complete regular bipartite graph.Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the AM

    Graph entropy and related topics

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