1,163 research outputs found

    Partition-crossing hypergraphs

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    For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . . , Xk) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Tur´an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs

    Partition-Crossing Hypergraphs

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    For a finite set XX, we say that a set HXH\subseteq X crosses a partition P=(X1,,Xk){\cal P}=(X_1,\dots,X_k) of XX if HH intersects min(H,k)\min (|H|,k) partition classes. If Hk|H|\geq k, this means that HH meets all classes XiX_i, whilst for Hk|H|\leq k the elements of the crossing set HH belong to mutually distinct classes. A set system H{\cal H} crosses P{\cal P}, if so does some HHH\in {\cal H}. The minimum number of rr-element subsets, such that every kk-partition of an nn-element set XX is crossed by at least one of them, is denoted by f(n,k,r)f(n,k,r). The problem of determining these minimum values for k=rk=r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387--1404]. The present authors determined asymptotically tight estimates on f(n,k,k)f(n,k,k) for every fixed kk as nn\to \infty [Graphs Combin., 25 (2009), 807--816]. Here we consider the more general problem for two parameters kk and rr, and establish lower and upper bounds for f(n,k,r)f(n,k,r). For various combinations of the three values n,k,rn,k,r we obtain asymptotically tight estimates, and also point out close connections of the function f(n,k,r)f(n,k,r) to Tur\'an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.Comment: 15 page

    Tur\'annical hypergraphs

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    This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.Comment: 33 pages, minor improvements thanks to two referee

    Counting and enumerating optimum cut sets for hypergraph kk-partitioning problems for fixed kk

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    We consider the problem of enumerating optimal solutions for two hypergraph kk-partitioning problems -- namely, Hypergraph-kk-Cut and Minmax-Hypergraph-kk-Partition. The input in hypergraph kk-partitioning problems is a hypergraph G=(V,E)G=(V, E) with positive hyperedge costs along with a fixed positive integer kk. The goal is to find a partition of VV into kk non-empty parts (V1,V2,,Vk)(V_1, V_2, \ldots, V_k) -- known as a kk-partition -- so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-kk-Partition. A subset of hyperedges is a minmax-kk-cut-set if it is the subset of hyperedges crossing an optimum kk-partition for Minmax-Hypergraph-kk-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the kk-partition, then the problem is known as Hypergraph-kk-Cut. A subset of hyperedges is a min-kk-cut-set if it is the subset of hyperedges crossing an optimum kk-partition for Hypergraph-kk-Cut. We give the first polynomial bound on the number of minmax-kk-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed kk. Our technique is strong enough to also enable an nO(k)pn^{O(k)}p-time deterministic algorithm to enumerate all min-kk-cut-sets in hypergraphs, thus improving on the previously known nO(k2)pn^{O(k^2)}p-time deterministic algorithm, where nn is the number of vertices and pp is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-kk-Cut and Minmax-Hypergraph-kk-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with arXiv:2110.1481

    Hamilton cycles in 5-connected line graphs

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    A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness
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