188 research outputs found

    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

    Stability for the Erdős-Rothschild problem

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    Given a sequence k:=(k1,,ks)\boldsymbol {k} := (k_1,\ldots ,k_s) of natural numbers and a graph G, let F(G;k)F(G;\boldsymbol {k}) denote the number of colourings of the edges of G with colours 1,,s1,\dots ,s , such that, for every c{1,,s}c \in \{1,\dots ,s\} , the edges of colour c contain no clique of order kck_c . Write F(n;k)F(n;\boldsymbol {k}) to denote the maximum of F(G;k)F(G;\boldsymbol {k}) over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/(n2)\log _2 F(n;\boldsymbol {k})/{n\choose 2} as n tends to infinity and proved a stability theorem for complete multipartite graphs G

    Hereditary classes of graphs : a parametric approach

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    The world of hereditary classes is rich and diverse and it contains a variety of classes of theoretical and practical importance. Thousands of results in the literature are devoted to individual classes and only a few of them analyse the universe of hereditary classes as a whole. To shift the analysis into a new level, in the present paper we exploit an approach, where we operate by infinite families of classes, rather than individual classes. Each family is associated with a graph parameter and is characterized by classes that are critical with respect to the parameter. In particular, we obtain a complete parametric description of the bottom of the lattice of hereditary classes and discuss a number of open questions related to this approach

    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs H=(H1,,Hm)\textbf{H}=(H_1,\ldots,H_m) with the same vertex set, an mm-edge graph Fi[m]HiF\subset \cup_{i\in [m]}H_i is a transversal if there is a bijection ϕ:E(F)[m]\phi:E(F)\to [m] such that eE(Hϕ(e))e\in E(H_{\phi(e)}) for each eE(F)e\in E(F). How large does the minimum degree of each HiH_i need to be so that H\textbf{H} necessarily contains a copy of FF that is a transversal? Each HiH_i in the collection could be the same hypergraph, hence the minimum degree of each HiH_i needs to be large enough to ensure that FHiF\subseteq H_i. Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020, 52(3):498-504], a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of rnrn graphs on an nn-vertex set, each with minimum degree at least (r/(r+1)+o(1))n(r/(r+1)+o(1))n, contains a transversal copy of the rr-th power of a Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour conjecture.Comment: 21 pages, 4 figures; final version as accepted for publication in the Bulletin of the London Mathematical Societ

    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

    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs =(1,...,) with the same vertex set, an -edge graph ⊂∪∈[] is atransversal if there is a bijection ∶()→[] such that ∈(()) for each ∈(). How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph,hence the minimum degree of each needs to be large enough to ensure that ⊆. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020)498–504), a growing body of work has shown that inmany cases this lower bound is tight. In this paper, wegive a unified approach to this problem by providinga widely applicable sufficient condition for this lowerbound to be asymptotically tight. This is general enoughto recover many previous results in the area and obtainnovel transversal variants of several classical Dirac-typeresults for (powers of) Hamilton cycles. For example, wederive that any collection of graphs on an -vertex set, each with minimum degree at least (∕( + 1) +(1)), contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow versionof the Pósa–Seymour conjecture

    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

    Smoothed Analysis of the Koml\'os Conjecture: Rademacher Noise

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    The {\em discrepancy} of a matrix MRd×nM \in \mathbb{R}^{d \times n} is given by DISC(M):=minx{1,1}nMx\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty. An outstanding conjecture, attributed to Koml\'os, stipulates that DISC(M)=O(1)\mathrm{DISC}(M) = O(1), whenever MM is a Koml\'os matrix, that is, whenever every column of MM lies within the unit sphere. Our main result asserts that DISC(M+R/d)=O(d1/2)\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2}) holds asymptotically almost surely, whenever MRd×nM \in \mathbb{R}^{d \times n} is Koml\'os, RRd×nR \in \mathbb{R}^{d \times n} is a Rademacher random matrix, d=ω(1)d = \omega(1), and n=ω~(d5/4)n = \tilde \omega(d^{5/4}). We conjecture that n=ω(dlogd)n = \omega(d \log d) suffices for the same assertion to hold. The factor d1/2d^{-1/2} normalising RR is essentially best possible.Comment: For version 2, the bound on the discrepancy is improve

    Transversals via regularity

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    Given graphs G1,,GsG_1,\ldots,G_s all on the same vertex set and a graph HH with e(H)se(H) \leq s, a copy of HH is transversal or rainbow if it contains at most one edge from each GcG_c. When s=e(H)s=e(H), such a copy contains exactly one edge from each GiG_i. We study the case when HH is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs HH. Our proofs use weak regularity in the 33-uniform hypergraph whose edges are those xycxyc where xyxy is an edge in the graph GcG_c. We apply our lemma to give a large class of spanning 33-uniform linear hypergraphs HH such that any sufficiently large uniformly dense nn-vertex 33-uniform hypergraph with minimum vertex degree Ω(n2)\Omega(n^2) contains HH as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft

    Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture

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    An orthomorphism of a finite group GG is a bijection ϕ ⁣:GG\phi\colon G\to G such that gg1ϕ(g)g\mapsto g^{-1}\phi(g) is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when GG is abelian, for any k2k\geq 2 dividing G1|G|-1, there exists an orthomorphism of GG fixing the identity and permuting the remaining elements as products of disjoint kk-cycles. We prove this conjecture for all sufficiently large groups.Comment: 34 page
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