Almost fair perfect matchings in complete bipartite graphs

Let {E₁,…,E_m} be a partition of E(K_n,n), where K_n,n is the complete bipartite graph, and assume that [Formula presented]. It was conjectured in [1], that there exists a perfect matching M in K_n,n with [Formula presented] In this paper, we reprove combinatorially that this conjecture is true when m=2 or m=3. This result is proved in [1] by using topological methods. In the case m=4, we prove that there is always a perfect matching M in K_n,n with s(M)≤11. We also bring here an unpublished result from 2014 of the second author of this paper together with Irine Lo and Paul Seymour, proving that there exists a function of m alone, f(m), and a perfect matching M in K_n,n such that s(M)≤f(m). This result was later reproved by Alon in [2], where an explicit formulation of f(m) was given.