On Simple Hamiltonian Cycles in a $2$-Colored Complete Graph

Abstract

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.