The complexity of vertex coloring problems in uniform hypergraphs with high degree
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Perfect matchings (and Hamilton cycles) in hypergraphs with large degrees
European Journal of Combinatorics
Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels
Journal of Combinatorial Theory Series A
Exact minimum degree thresholds for perfect matchings in uniform hypergraphs
Journal of Combinatorial Theory Series A
European Journal of Combinatorics
Matchings in 3-uniform hypergraphs
Journal of Combinatorial Theory Series B
Fractional and integer matchings in uniform hypergraphs
European Journal of Combinatorics
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For a k-graph F, let t l (n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of vertex-disjoint F-subgraphs covering all but at most m vertices of G. Let K m k denote the complete k-graph on m vertices. The function $$t_{k-1} (kn, 0, K_k^k)$$(i.e. when we want to guarantee a perfect matching) has been previously determined by Kühn and Osthus [9] (asymptotically) and by Rödl, Ruciński, and Szemerédi [13] (exactly). Here we obtain asymptotic formulae for some other l. Namely, we prove that for any $$k \ge 4$$and $$k/2 \le l \le k-2$$, $$ t_l(kn, 0, K_k^k) = \left(\frac{1}{2} + o(1)\right) {kn\choose k-l} $$is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that $${\rm sd}_{\gamma} (G) \le \delta(G) + 1$$for any graph G with $$\delta(G) \ge 2$$. In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.