不带零件的包装
Packing without some pieces
摘要：
Erdés和Hanani证明了，对于每一个固定的整数$k \ge 2$，完全图$K_n$几乎可以被$K_k$的副本完全填充;也就是说，$K_n$包含$K_k$的成对边不相交的副本，这些副本覆盖了它的所有边，除了$o_n(1)$的一部分。等价地，$K_k$的所有红-蓝边着色的集合$\C(k)$的元素可以用于几乎完全包装$K_n$的每个红-蓝边着色。考虑上述Erdens-Hanani结果的以下加强。假设$\C' \subset \C(k)$。我们真的可以只使用$\C'$中的元素，并且几乎完全包装$K_n$的每一个红-蓝边着色吗？如果$\C'=\C(k) \setminus C$具有此属性，则元素$C \in \C(k)$是{ \em可避免}，并且如果$\C'=\C(k) \setminus {\cal F}$具有此属性，则子集${\cal F} \subset \C(k)$是可避免的。似乎很难确定所有可避免的图以及所有可避免的族。我们证明了一些非平凡的充分条件的可避免性。我们的证明意味着，特别是，（i）几乎所有的元素的$\C(k)$是可避免的（ii）所有欧拉元素的$\C(k)$是可避免的，事实上，所有欧拉元素的集合的$\C(k)$是可避免的。
英文摘要：
Erdős 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 Erdős-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.
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