Almost color-balanced perfect matchings in color-balanced complete graphs

Abstract

For a graph G and a not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges of G of color i, and call G color-balanced if mi(G)=mj(G) for every two colors i and j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly n-edge colored complete bipartite graph Kn,n has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M. For a perfect matching M of K2kn, a natural measure for the total deviation of M from being color-balanced is f(M)=∑i=1k|mi(M)−n|. While not every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M, that is, a perfect matching with f(M)=0, we prove the existence of a perfect matching M with f(M)=O(kknln⁡(k)) for general k and f(M)≤2 for k=3; the case k=2 has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.