A Conjecture on Rainbow Hamiltonian Cycle Decomposition

Wu in 1999 conjectured that if H is a subgraph of the complete graph K₂ₙ₊₁ with n edges, then there is a Hamiltonian cycle decomposition of K₂ₙ₊₁ such that each edge of H is in a separate Hamiltonian cycle. The conjecture was partially settled by Liu and Chen (2023) in cases that |V(H)|≤ n+1, H is a linear forest, or n≤ 5. In this paper, we settle the conjecture completely. This result can be viewed as a complete graph analogous of Evans conjecture and has some applications in linear arboricity conjecture and restricted size Ramsey numbers.