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# Almost fair perfect matchings in complete bipartite graphs

DISCRETE MATHEMATICS（2024）

Let {E-1, ..., E-m} be a partition of E(Kn,n), where Kn,n is the complete bipartite graph, and assume that |E-i|/n is an element of Z. It was conjectured in [1], that there exists a perfect matching M in K-n,K-n with s(M) = max(f) (|M boolean AND Ei|/n ) - min (i)(|M boolean AND Ei| - |Ei|/n )<= 2 . 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 Kn,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 We also bring here an unpublished result from 2014 of the second author of this paper together with Irine Lo and Paul Seymour, proving that there exists a function of m alone, f (m), and a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 Kn,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. (c) 2023 Elsevier B.V. All rights reserved.

Graphs matrices,Perfect-matchings,Latin -squares
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