Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions

Details Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions

2003-11-25

arxiv.org/abs/math/0311449v1

Mathematics

Combinatorics

12 pages

Scientific paper

Let $H$ be a fixed graph. A {\em 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 $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-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 $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges.

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