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1.
If T is an n‐vertex tournament with a given number of 3‐cycles, what can be said about the number of its 4‐cycles? The most interesting range of this problem is where T is assumed to have cyclic triples for some and we seek to minimize the number of 4‐cycles. We conjecture that the (asymptotic) minimizing T is a random blow‐up of a constant‐sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of 4‐cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in T, how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments. 相似文献
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《Journal of Graph Theory》2018,87(4):399-429
We consider an extremal problem motivated by a article of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge‐colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given , we look for n‐vertex graphs that admit the maximum number of r‐edge‐colorings such that at most colors appear in edges incident with each vertex, that is, r‐edge‐colorings avoiding rainbow‐colored stars with t edges. For large n, we show that, with the exception of the case , the complete graph is always the unique extremal graph. We also consider generalizations of this problem. 相似文献
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Petra Johann 《Journal of Graph Theory》2000,35(4):227-243
In this paper we study the structure of graphs with a unique k‐factor. Our results imply a conjecture of Hendry on the maximal number m (n,k) of edges in a graph G of order n with a unique k‐factor: For we prove and construct all corresponding extremal graphs. For we prove . For n = 2kl, l ∈ ℕ, this bound is sharp, and we prove that the corresponding extremal graph is unique up to isomorphism. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 227–243, 2000 相似文献
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Dirk Meierling 《Journal of Graph Theory》2010,63(1):82-92
A directed cycle C of a digraph D is extendable if there exists a directed cycle C′ in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non‐Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3≤k<n: a digraph D is k ‐extendable if every directed cycle of length t, where k≤t<n, is extendable. Moreover, D is called fully k ‐extendable if D is k ‐extendable and every vertex of D belongs to a directed cycle of length k. An in‐tournament is an oriented graph such that the in‐neighborhood of every vertex induces a tournament. This class of digraphs which generalizes the class of tournaments was introduced by Bang‐Jensen, Huang and Prisner in 1993. Tewes and Volkmann showed that every connected in‐tournament D of order n with minimum degree δ≥1 is ( ) ‐extendable. Furthermore, if D is a strongly connected in‐tournament of order n with minimum degree δ=2 or , then D is fully ( ) ‐extendable. In this article we shall see that if , every vertex of D belongs to a directed cycle of length , which means that D is fully ( ) ‐extendable. This confirms a conjecture of Tewes and Volkmann. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 82–92, 2010 相似文献
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We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd?s, and Sós implies that every n‐vertex graph with odd girth and minimum degree bigger than must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases and 3, we show that every n‐vertex graph with odd girth and minimum degree bigger than is homomorphic to the cycle of length . This is best possible in the sense that there are graphs with minimum degree and odd girth that are not homomorphic to the cycle of length . Similar results were obtained by Brandt and Ribe‐Baumann. 相似文献
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Barnaby Roberts 《Journal of Graph Theory》2017,85(2):429-445
We look at several saturation problems in complete balanced blow‐ups of graphs. We let denote the blow‐up of H onto parts of size n and refer to a copy of H in as partite if it has one vertex in each part of . We then ask how few edges a subgraph G of can have such that G has no partite copy of H but such that the addition of any new edge from creates a partite H. When H is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger in 5 . Our main result is to calculate this value for when n is large. We also give exact results for paths and stars and show that for 2‐connected graphs the answer is linear in n whilst for graphs that are not 2‐connected the answer is quadratic in n. We also investigate a similar problem where G is permitted to contain partite copies of H but we require that the addition of any new edge from creates an extra partite copy of H. This problem turns out to be much simpler and we attain exact answers for all cliques and trees. 相似文献
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Jürgen Kritschgau Abhishek Methuku Michael Tait Craig Timmons 《Journal of Graph Theory》2020,94(3):320-348
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An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k‐pancyclicity of in‐tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in‐tournament is vertex k‐pancyclic for k ≤ 5 and k ≥ n − 3. In the latter case, we even show that the in‐tournaments in consideration are fully (n − 3)‐extendable which means that every vertex belongs to a cycle of length n − 3 and that the vertex set of every cycle of length at least n − 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in‐tournament of order n with minimum degree greater than is vertex k‐pancyclic for 5 < k < n − 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001 相似文献
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Tomoki Nakamigawa 《Journal of Graph Theory》2007,56(3):159-166
Let k be a fixed integer at least 3. It is proved that every graph of order (2k ? 1 ? 1/k)n + O(1) contains n vertex disjoint induced subgraphs of order k such that these subgraphs are equivalent to each other and they are equivalent to one of four graphs: a clique, an independent set, a star, or the complement of a star. In particular, by substituting 3 for k, it is proved that every graph of order 14n/3 + O(1) contains n vertex disjoint induced subgraphs of order 3 such that they are equivalent to each other. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 159–166, 2007 相似文献
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Dhruv Mubayi 《Journal of Graph Theory》2012,70(2):171-179
A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ?n2/4?; + 1 edges. Rademacher proved that G contains at least ?n/2?; triangles, and several authors proved that G contains a book of size at least n/6. We prove the following “linear combination” of these two results. Suppose that and the maximum size of a book in G is less than αn/2. Then G contains at least triangles as n→?. This is asymptotically sharp. On the other hand, for every , there exists β>0 such that G contains at least βn3 triangles. It remains an open problem to determine the largest possible β in terms of α. Our short proof uses the triangle removal lemma, although there is another approach which avoids this. © 2011 Wiley Periodicals, Inc. J Graph Theory 相似文献
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《Journal of Graph Theory》2018,88(1):192-210
A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W4 and L4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments with is called almost balanced if all but vertices of have outdegree . In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W4 and L4 in goes to zero as n goes to infinity. 相似文献
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Let be a graph, be an integer, and write for the maximum number of edges in an ‐vertex graph that is ‐partite and has no subgraph isomorphic to . The function has been studied by many researchers. Finding is a special case of the Zarankiewicz problem. We prove an analog of the Kövári‐Sós‐Turán theorem for 3‐partite graphs by showing for . Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case that and is odd, that is, for . In the cases of and , we use a result of Allen, Keevash, Sudakov, and Verstraëte, to show that a similar upper bound holds for all and gives a better constant when . Finally, we point out an interesting connection between difference families from design theory and . 相似文献
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《Journal of Graph Theory》2018,88(3):411-427
Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define Erdős, Łuczak, and Spencer proved that for , In this article, we prove the following lower bound: for , Furthermore, we show that this is tight up to a multiplicative constant factor for infinitely many p near the elements of . In contrast, we show that for any n‐vertex graph G, either G or contains a full subgraph on vertices. Finally, we discuss full subgraphs of random and pseudo‐random graphs, and several open problems. 相似文献
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Maya Jakobine Stein 《Journal of Graph Theory》2007,54(4):331-349
A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007 相似文献
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《组合设计杂志》2018,26(10):487-504
Given a set S of symbols, and integers and , an array is said to cover a t‐set of columns if all sequences in appear as rows in the corresponding subarray of A. If A covers all t‐subsets of columns, it is called an ‐covering array. These arrays have a wide variety of applications, driving the search for small covering arrays. Here, we consider an inverse problem: rather than aiming to cover all t‐sets of columns with the smallest possible array, we fix the size N of the array to be equal to and try to maximize the number of covered t‐sets. With the machinery of hypergraph Lagrangians, we provide an upper bound on the number of t‐sets that can be covered. A linear algebraic construction shows this bound to be tight; exactly so in the case when v is a prime power and divides k, and asymptotically so in other cases. As an application, by combining our construction with probabilistic arguments, we match the best‐known asymptotics of the covering array number , which is the smallest N for which an ‐covering array exists, and improve the upper bounds on the almost‐covering array number . 相似文献
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Graham Brightwell Konstantinos Panagiotou Angelika Steger 《Random Structures and Algorithms》2012,41(2):147-178
We prove that there is a constant c > 0, such that whenever p ≥ n‐c, with probability tending to 1 when n goes to infinity, every maximum triangle‐free subgraph of the random graph Gn,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599–622). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M ? n and M ≤ /2, is “nearly unique”. More precisely, given a maximum cut C of Gn,M, we can obtain all maximum cuts by moving at most begin{align*}mathcal{O}(sqrt{n^3/M})end{align*} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 相似文献
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If x is a vertex of a digraph D, then we denote by d
+(x) and d
−(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p ∈ ℕ such that d
+(x) = d
−(x) = p for all vertices x of D.
A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a
given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether
c-partite tournaments with r vertices in each partite set contain a cycle with exactly r − 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c ⩾ 3, then we will show that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a cycle with exactly r − 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2. 相似文献
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Komlós [Komlós: Tiling Turán Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal–Szemerédi theorem that deals with the case when H is a clique. We give a proof of a graphon version of Komlós's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladký, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlós's theorem. 相似文献