On Simple Hamiltonian Cycles in a \(2\)-Colored Complete Graph

Let the edges of the complete graph \(K_n\) be \(2\)-colored. A Simple Hamiltonian Cycle is a Hamiltonian cycle in \(K_n\) that is either monochromatic or is a union of two monochromatic paths. The main result of this paper is that if \(n\) is an even integer greater than \(4\), then for every \(2\)-coloring of the edges of \(K_n\), there is a Simple Hamiltonian Cycle in \(K_n\) which is either monochromatic, or is a union of two monochromatic paths, where each path is of even length.