  Motivated by the problem in [6], which studies the relative efficiency of
propositional proof systems, 2-edge colorings of complete bipartite graphs are
investigated. It is shown that if the edges of $G=K_{n,n}$ are colored with
black and white such that the number of black edges differs from the number of
white edges by at most 1, then there are at least $n(1-1/\sqrt{2})$
vertex-disjoint forks with centers in the same partite set of $G$. Here, a fork
is a graph formed by two adjacent edges of different colors. The bound is
sharp. Moreover, an algorithm running in time $O(n^2 \log n \sqrt{n
\alpha(n^2,n) \log n})$ and giving a largest such fork forest is found.
