On Integer Balancing of Digraphs

TL;DR

This paper proves that in weighted directed graphs with integer in-degree and out-degree sums at each vertex, the edge weights can also be chosen to be integers, ensuring a balanced configuration.

Contribution

It establishes that integer degree sums in a weighted digraph guarantee the existence of integer edge weights for balancing.

Findings

Integer degree sums imply the existence of integer edge weights for balance.

The result applies to weighted digraphs with integer in-degree and out-degree sums.

Provides a theoretical foundation for integer balancing in directed graphs.

Abstract

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 the edge weights can be integers as well.