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A Conjecture on Rainbow Hamiltonian Cycle Decomposition

Ramin Javadi, Meysam Miralaei

Abstract

Wu in 1999 conjectured that if $H$ is a subgraph of the complete graph $K_{2n+1}$ with $n$ edges, then there is a Hamiltonian cycle decomposition of $K_{2n+1}$ 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 $|V(H)|\leq n+1$, $H$ is a linear forest, or $n\leq 5$. 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.

A Conjecture on Rainbow Hamiltonian Cycle Decomposition

Abstract

Wu in 1999 conjectured that if is a subgraph of the complete graph with edges, then there is a Hamiltonian cycle decomposition of such that each edge of is in a separate Hamiltonian cycle. The conjecture was partially settled by Liu and Chen (2023) in cases that , is a linear forest, or . 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.
Paper Structure (6 sections, 10 theorems, 3 equations)

This paper contains 6 sections, 10 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Notations and Terminologies
  3. Embedding Components Non-isomorphic to $K_2$
  4. Embedding Components Isomorphic to $K_2$
  5. Concluding Remarks
  6. Proof of Lemma \ref{['lem:hilton']}

Key Result

Theorem 2

liu-chen Suppose that $H$ is a subgraph of $K_{2n+1}$ with $n$ edges such that either Then, there is a Hamiltonian cycle decomposition $\mathcal{C}$ of $K_{2n+1}$ such that $H$ is rainbow in $\mathcal{C}$.

Theorems & Definitions (12)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 2 more