Fork-forests in bi-colored complete bipartite graphs

TL;DR

This paper investigates 2-edge colorings of complete bipartite graphs, establishing a sharp lower bound on the number of vertex-disjoint forks with centers in the same part, and provides an efficient algorithm to find the largest such fork forest.

Contribution

It introduces a new bound on the number of vertex-disjoint forks in bi-colored complete bipartite graphs and presents an algorithm to find the maximum fork forest efficiently.

Findings

At least n(1-1/√2) vertex-disjoint forks exist under certain coloring conditions.

The bound on the number of forks is proven to be sharp.

An O(n^2 log n √(n α(n^2,n) log n)) algorithm for finding the largest fork forest is developed.

Abstract

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.