A note on equitable Hamiltonian cycles

Given a complete graph with an even number of vertices, and with each edge colored with one of two colors (say red or blue), an equitable Hamiltonian cycle is a Hamiltonian cycle that can be decomposed into two perfect matchings such that both perfect matchings have the same number of red edges. We show that, for any coloring of the edges, in any complete graph on at least 6 vertices, an equitable Hamiltonian cycle exists.