Integer and fractional packing of families of graphs

Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $\nu_{{\cal F}}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $\psi$ from the set of elements of ${\cal F}$ in $G$ to $[0,1]$ is a {\em fractional ${\cal F}$-packing} of $G$ if $\sum_{e \in H \in {\cal F}} {\psi(H)} \leq 1$ for each $e \in E(G)$. The {\em fractional ${\cal F}$-packing number}, denoted $\nu^*_{{\cal F}}(G)$, is defined to be the maximum value of $\sum_{H \in {{G} \choose {{\cal F}}}} \psi(H)$ over all fractional ${\cal F}$-packings $\psi$. Our main result is that $\nu^*_{{\cal F}}(G)-\nu_{{\cal F}}(G) = o(|V(G)|^2)$. Furthermore, a set of $\nu_{{\cal F}}(G) -o(|V(G)|^2)$ edge-disjoint elements of ${\cal F}$ in $G$ can be found in randomized polynomial time. For the special case ${\cal F}=\{H_0\}$ we obtain a significantly simpler proof of a recent difficult result of Haxell and R\"odl \cite{HaRo} that $\nu^*_{H_0}(G)-\nu_{H_0}(G) = o(|V(G)|^2)$.