381 research outputs found

    Covering and Separation for Permutations and Graphs

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    This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph

    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

    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

    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

    Influence Maximization based on Simplicial Contagion Model in Hypergraph

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    In recent years, the issue of node centrality has been actively and extensively explored due to its applications in product recommendations, opinion propagation, disease spread, and other areas involving maximizing node influence. This paper focuses on the problem of influence maximization on the Simplicial Contagion Model, using the susceptible-infectedrecovered (SIR) model as an example. To find practical solutions to this optimization problem, we have developed a theoretical framework based on message passing processes and conducted stability analysis of equilibrium solutions for the self-consistent equations. Furthermore, we introduce a metric called collective influence and propose an adaptive algorithm, known as the Collective Influence Adaptive (CIA), to identify influential propagators in the spreading process. This method has been validated on both synthetic hypergraphs and real hypergraphs, outperforming other competing heuristic methods.Comment: 19 pages,16 figure

    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
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