A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums are integers, then for any edge weights satisfying the balance conditions, there exist integer weights obtained by rounding the original weights up or down that preserve both the balance condition and the sum of all edge weights.

中文翻译：

---

关于有向图的整数平衡

如果每个顶点的入边和出边的权重之和相等，则加权有向图是平衡的。我们表明，如果这些总和是整数，那么对于满足平衡条件的任何边权重，存在通过向上或向下舍入原始权重获得的整数权重，同时保留平衡条件和所有边权重的总和。