On perfect packings in dense graphs

Details On perfect packings in dense graphs On perfect packings in dense graphs

2011-10-16

arxiv.org/abs/1110.3490v1

Mathematics

Combinatorics

19 pages, 1 figure

Scientific paper

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect H-packings: Given positive intergers n, r, D, we characterise the edge density threshold that ensures a perfect K_r-packing in any graph G on n vertices and with minimum degree at least D. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect H-packing. Other related embedding problems are also considered. Indeed, we give a degree sequence condition which forces a graph to contain a copy of K_r, thereby strengthening the minimum degree version of Turan's theorem. We also characterise the edge density threshold that ensures a graph G contains k vertex-disjoint cycles.

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