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61.
József Balogh Jane Butterfield Ping Hu John Lenz 《Random Structures and Algorithms》2016,48(4):641-654
A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle‐free subgraph of Kn is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large p, every maximum triangle‐free subgraph of G(n, p) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for for some constant K, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by F5, which is sometimes called the generalized triangle, the 3‐uniform hypergraph with vertex set and edge set . One of the first results in extremal hypergraph theory is by Frankl and Füredi, who proved that the maximum 3‐uniform hypergraph on n vertices containing no copy of F5 is tripartite for n > 3000. A natural question is for what p is every maximum F5‐free subhypergraph of w.h.p. tripartite. We show this holds for for some constant K and does not hold for . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 641–654, 2016 相似文献
62.
In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C?(r, t), as a sequence of distinct vertices v1, v2, … , v?, such that for each set (vi, vi + 1, … , vi + t ? 1) of t consecutive vertices on the cycle, there is an edge Ei of that contains these t vertices and the edges Ei are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of Kn(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008 相似文献
63.
Beka Ergemlidze 《Discrete Mathematics》2021,344(4):112262
In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge 4-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound. 相似文献
64.
Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP‐hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP‐hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial‐time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008 相似文献
65.
Counting acyclic hypergraphs 总被引:4,自引:0,他引:4
Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct
acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the
explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs. 相似文献
66.
For a hypergraph G and a positive integer s, let be the minimum value of l such that G is L‐colorable from every list L with for each and for all . This parameter was studied by Kratochvíl, Tuza, and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of for balanced complete multipartite graphs and for complete k‐partite k‐uniform hypergraphs. 相似文献
67.
The well‐known Ramsey number is the smallest integer n such that every ‐free graph of order n contains an independent set of size u. In other words, it contains a subset of u vertices with no K2. Erd?s and Rogers introduced a more general problem replacing K2 by for . Extending the problem of determining Ramsey numbers they defined the numbers where the minimum is taken over all ‐free graphs G of order n. In this note, we study an analogous function for 3‐uniform hypergraphs. In particular, we show that there are constants c1 and c2 depending only on s such that 相似文献
68.
69.
Dmitry A. Shabanov 《Random Structures and Algorithms》2012,40(2):227-253
The work deals with a combinatorial problem of P. Erd?s and L. Lovász concerning simple hypergraphs. Let denote the minimum number of edges in an n‐uniform simple hypergraph with chromatic number at least . The main result of the work is a new asymptotic lower bound for . We prove that for large n and r satisfying the following inequality holds where . This bound improves previously known bounds for . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h‐simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 相似文献
70.
A mixed hypergraph is a triple H=(X,C,D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c:X→[k] such that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle.We suggest a general procedure for coloring circular mixed hypergraphs and prove that if H is a reduced colorable circular mixed hypergraph with n vertices, upper chromatic number and sieve number s, then