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Daniela Kühn

Publications and source records attributed to Daniela Kühn.

81 records · Page 5Linked to original sources
arXiv2008

Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.

math.CO↗
arXiv2008

A Dirac type result on Hamilton cycles in oriented graphs

We show that for each α>0 every sufficiently large oriented graph G with δ^+(G),δ^-(G)\ge 3|G|/8+ α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if δ(G)+δ^+(G)+δ^-(G)\geq 3|G|/2 + α|G|. Up to the term α|G| this confirms a conjecture of Häggkvist. We also prove an Ore-type theorem for oriented graphs.

math.CO↗
arXiv2008

The minimum degree threshold for perfect graph packings

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let delta(H,n) denote the smallest integer t such that every graph G whose order n is divisible by |H| and with delta(G) > t contains a perfect H-packing. We show that delta(H,n) = (1-1/χ*(H))n+O(1). The value of chi*(H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies k-1 < chi*(H) \le k, where k is the chromatic number of H.

math.CO↗
arXiv2007

k-Ordered Hamilton cycles in digraphs

Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sárközy and Selkow.

math.CO↗
arXiv2007

Linkedness and ordered cycles in digraphs

The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order.

math.CO↗
arXiv2006

3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

Chvátal, Rödl, Szemerédi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform `pseudo-random' hypergraphs.

math.CO↗
arXiv2006

Perfect packings with complete graphs minus an edge

Let K_r^- denote the graph obtained from K_r by deleting one edge. We show that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a collection of disjoint copies of K_r^- which covers all vertices of G. Here chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n.

math.CO↗
arXiv2006

A note on complete subdivisions in digraphs of large outdegree

Mader conjectured that for all k there is an integer d(k) such that every digraph of minimum outdegree at least d(k) contains a subdivision of a transitive tournament of order k. In this note we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let G be a digraph of order n whose minimum outdegree is at least d. Then G contains a subdivision of a complete digraph of order at least d^2/(8n^{3/2}).

math.CO↗