651 research outputs found

    On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

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    The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.Comment: 17 pages, 5 figure

    An extension of Tur\'an's Theorem, uniqueness and stability

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    We determine the maximum number of edges of an nn-vertex graph GG with the property that none of its rr-cliques intersects a fixed set MV(G)M\subset V(G). For (r1)Mn(r-1)|M|\ge n, the (r1)(r-1)-partite Turan graph turns out to be the unique extremal graph. For (r1)M<n(r-1)|M|<n, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's comments incorporate

    On the Tur\'an number of the hypercube

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    In 1964, Erd\H{o}s proposed the problem of estimating the Tur\'an number of the dd-dimensional hypercube QdQ_d. Since QdQ_d is a bipartite graph with maximum degree dd, it follows from results of F\"uredi and Alon, Krivelevich, Sudakov that ex(n,Qd)=Od(n21/d)\mathrm{ex}(n,Q_d)=O_d(n^{2-1/d}). A recent general result of Sudakov and Tomon implies the slightly stronger bound ex(n,Qd)=o(n21/d)\mathrm{ex}(n,Q_d)=o(n^{2-1/d}). We obtain the first power-improvement for this old problem by showing that ex(n,Qd)=Od(n21d1+1(d1)2d1)\mathrm{ex}(n,Q_d)=O_d(n^{2-\frac{1}{d-1}+\frac{1}{(d-1)2^{d-1}}}). This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any nn-vertex, properly edge-coloured graph without a rainbow cycle has at most O(n(logn)2)O(n(\log n)^2) edges, improving the previous best bound of n(logn)2+o(1)n(\log n)^{2+o(1)} by Tomon. Furthermore, we show that any properly edge-coloured nn-vertex graph with ω(nlogn)\omega(n\log n) edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.Comment: 19 page

    An extension of Turán's theorem, uniqueness and stability

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    We determine the maximum number of edges of an n -vertex graph G with the property that none of its r -cliques intersects a fixed set M⊂V(G) . For (r−1)|M|≥n , the (r−1) -partite Turán graph turns out to be the unique extremal graph. For (r−1)|M|<n , there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results

    On the Chromatic Thresholds of Hypergraphs

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    Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c(V(H)r1)c \binom{|V(H)|}{r-1} has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure

    On a Tree and a Path with no Geometric Simultaneous Embedding

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    Two graphs G1=(V,E1)G_1=(V,E_1) and G2=(V,E2)G_2=(V,E_2) admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for G1G_1 and for G2G_2. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.Comment: 42 pages, 33 figure

    Bounding sequence extremal functions with formations

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    An (r,s)(r, s)-formation is a concatenation of ss permutations of rr letters. If uu is a sequence with rr distinct letters, then let Ex(u,n)\mathit{Ex}(u, n) be the maximum length of any rr-sparse sequence with nn distinct letters which has no subsequence isomorphic to uu. For every sequence uu define fw(u)\mathit{fw}(u), the formation width of uu, to be the minimum ss for which there exists rr such that there is a subsequence isomorphic to uu in every (r,s)(r, s)-formation. We use fw(u)\mathit{fw}(u) to prove upper bounds on Ex(u,n)\mathit{Ex}(u, n) for sequences uu such that uu contains an alternation with the same formation width as uu. We generalize Nivasch's bounds on Ex((ab)t,n)\mathit{Ex}((ab)^{t}, n) by showing that fw((12l)t)=2t1\mathit{fw}((12 \ldots l)^{t})=2t-1 and Ex((12l)t,n)=n21(t2)!α(n)t2±O(α(n)t3)\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})} for every l2l \geq 2 and t3t\geq 3, such that α(n)\alpha(n) denotes the inverse Ackermann function. Upper bounds on Ex((12l)t,n)\mathit{Ex}((12 \ldots l)^{t} , n) have been used in other papers to bound the maximum number of edges in kk-quasiplanar graphs on nn vertices with no pair of edges intersecting in more than O(1)O(1) points. If uu is any sequence of the form avavaa v a v' a such that aa is a letter, vv is a nonempty sequence excluding aa with no repeated letters and vv' is obtained from vv by only moving the first letter of vv to another place in vv, then we show that fw(u)=4\mathit{fw}(u)=4 and Ex(u,n)=Θ(nα(n))\mathit{Ex}(u, n) =\Theta(n\alpha(n)). Furthermore we prove that fw(abc(acb)t)=2t+1\mathit{fw}(abc(acb)^{t})=2t+1 and Ex(abc(acb)t,n)=n21(t1)!α(n)t1±O(α(n)t2)\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})} for every t2t\geq 2.Comment: 25 page

    Rainbow Turán Problems

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    For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude

    Tilings in randomly perturbed dense graphs

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    A perfect HH-tiling in a graph GG is a collection of vertex-disjoint copies of a graph HH in GG that together cover all the vertices in GG. In this paper we investigate perfect HH-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds mm random edges to it. Specifically, for any fixed graph HH, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect HH-tiling with high probability. Our proof utilises Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect HH-tilings in dense graphs.Comment: 19 pages, to appear in CP
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