Rainbow subgraphs in Hamiltonian cycle decompositions of complete graphs

Abstract

A Hamiltonian cycle decomposition of a complete graph K2n+1 is an n-edge-coloring of K2n+1 such that each color class is a Hamiltonian cycle. For a given Hamiltonian cycle decomposition, a subgraph of K2n+1 is called rainbow if all its edges are colored differently. Let H be any subgraph of K2n+1 consisting of n edges. In 1999, Wu conjectured that K2n+1 has a Hamiltonian cycle decomposition such that H is rainbow. In this paper, we show that the conjecture is true in the cases that |H|≤n+1, or H is a linear forest, or n≤5. Using this result, we partially confirm a conjecture on restricted size Ramsey numbers due to Miralaei and Shahsiah in 2020.