Let $f(n,k)$ be the minimum size of a collection of bicliques such that (i) every edge of the complete graph $K_n$ is covered by at least one and at most $k$ bicliques in the collection, and (ii) for each edge $\{u,v\}$, the number of bicliques in which $u$ appears in the first class and $v$ in the second class differs by at most one from the number of bicliques in which $u$ appears in the second class and $v$ in the first class. For $k=1$, $f(n,k)$ reduces to the biclique partition number of $K_n$, and the Graham-Pollak theorem gives $f(n,1)=n-1$. For $k=2$, $f(n,k)$ is the ordered biclique partition number of $K_n$, for which it is known that $c_1 n^{1/2} \le f(n,2) \le c_2 n^{1/2+o(1)}$ for some positive constants $c_1$ and $c_2$. In this note, we give almost tight bounds for $f(n,k)$ for fixed $k \ge 2$: \[ (1+o(1))c_1(k)\cdot n^{\frac{1}{\lceil k/2\rceil+1}} \le f(n,k) \le (1+o(1))c_2(k)\cdot n^{\frac{1}{\lfloor k/2\rfloor+1}+o(1)}, \] where $c_1(k)$ and $c_2(k)$ are positive constants.

翻译：设$f(n,k)$表示满足以下条件的二分图族的最小规模：(i) 完全图$K_n$的每条边至少被族中一个且至多$k$个二分图覆盖；(ii) 对每条边$\{u,v\}$，$u$出现在第一类而$v$出现在第二类中的二分图个数，与$u$出现在第二类而$v$出现在第一类中的二分图个数相差至多为1。对于$k=1$，$f(n,k)$退化为$K_n$的二分图划分数，Graham-Pollak定理给出$f(n,1)=n-1$。对于$k=2$，$f(n,k)$是$K_n$的有序二分图划分数，已知存在正常数$c_1$和$c_2$使得$c_1 n^{1/2} \le f(n,2) \le c_2 n^{1/2+o(1)}$。本文中，我们对固定$k \ge 2$时的$f(n,k)$给出几乎紧的界： \[ (1+o(1))c_1(k)\cdot n^{\frac{1}{\lceil k/2\rceil+1}} \le f(n,k) \le (1+o(1))c_2(k)\cdot n^{\frac{1}{\lfloor k/2\rfloor+1}+o(1)}, \] 其中$c_1(k)$和$c_2(k)$为正常数。