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

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