Packing without some pieces

Abstract

Erd\H{o}s and Hanani proved that for every fixed integer $k \ge 2$, the complete graph $K_n$ can be almost completely packed with copies of $K_k$; that is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an $o_n(1)$ fraction of its edges. Equivalently, elements of the set $\C(k)$ of all red-blue edge colorings of $K_k$ can be used to almost completely pack every red-blue edge coloring of $K_n$. The following strengthening of the aforementioned Erd\H{o}s-Hanani result is considered. Suppose $\C' \subset \C(k)$. Is it true that we can use elements only from $\C'$ and almost completely pack every red-blue edge coloring of $K_n$? An element $C \in \C(k)$ is {\em avoidable} if $\C'=\C(k) \setminus C$ has this property and a subset ${\cal F} \subset \C(k)$ is avoidable if $\C'=\C(k) \setminus {\cal F}$ has this property. It seems difficult to determine all avoidable graphs as well as all avoidable families. We prove some nontrivial sufficient conditions for avoidability. Our proofs imply, in particular, that (i) almost all elements of $\C(k)$ are avoidable (ii) all Eulerian elements of $\C(k)$ are avoidable and, in fact, the set of all Eulerian elements of $\C(k)$ is avoidable.