Perfect Matchings and K 4 3 -Tilings in Hypergraphs of Large Codegree

Abstract

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 Kuhn and Osthus [9] (asymptotically) and by Rodl, Rucinski, and Szemeredi [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.