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# Asymptotically optimal Kk-packings of dense graphs via fractional Kk-decompositions

Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel
Journal of Combinatorial Theory, Series B DOI:10.1016/j.jctb.2005.02.002 pp.1-11
Source: arXiv

ABSTRACT Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of nonnegative real weights to the copies of H in G such that for each e∈E(G), the sum of the weights of copies of H containing e is precisely one. An H-packing of a graph G is a set of edge disjoint copies of H in G. The following results are proved. For every fixed k>2, every graph with n vertices and minimum degree at least n(1-1/9k10)+o(n) has a fractional Kk-decomposition and has a Kk-packing which covers all but o(n2) edges.

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##### Article:Fractional decompositions of dense hypergraphs
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ABSTRACT: A seminal result of Rödl (the Rödl nibble) asserts that the edges of the complete r -uniform hypergraph K n r can be packed, almost completely, with copies of K k r , where k is fixed. We prove that the same result holds in a dense hypergraph setting. It is shown that for every r -uniform hypergraph H 0 , there exists a constant α=α( H 0 )<1 such that every r -uniform hypergraph H in which every ( r −1)-set is contained in at least α n edges has an H 0 -packing that covers | E ( H )|(1− o n (1)) edges. Our method of proof uses fractional decompositions and makes extensive use of probabilistic arguments and additional combinatorial ideas.

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### Keywords

edge disjoint copies

fixed k>2

following results

fractional H-decomposition

fractional Kk-decomposition

H-packing

minimum degree

nonnegative real weights

weights