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A constructive solution to the Oberwolfach Problem with a large cycle

Tommaso Traetta

Abstract

For every $2$-regular graph $F$ of order $v$, the Oberwolfach problem $OP(F)$ asks whether there is a $2$-factorization of $K_v$ ($v$ odd) or $K_v$ minus a $1$-factor ($v$ even) into copies of $F$. Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to $OP(F)$ whenever $F$ contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building $2$-factorizations with an automorphism group having a nearly-regular action on the vertex-set.

A constructive solution to the Oberwolfach Problem with a large cycle

Abstract

For every -regular graph of order , the Oberwolfach problem asks whether there is a -factorization of ( odd) or minus a -factor ( even) into copies of . Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to whenever contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building -factorizations with an automorphism group having a nearly-regular action on the vertex-set.
Paper Structure (6 sections, 8 theorems, 33 equations)

This paper contains 6 sections, 8 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extending $(1,2)$-decompositions to $2$-factorizations
  4. Solutions to OP satisfying the matching property
  5. The proof of Theorem \ref{['main']}
  6. Conclusions

Key Result

Theorem 1.1

$OP(y,\ell_1, \ell_2, \ldots, \ell_u)$ has an explicit solution whenever where $b=\sum_{i=1}^u \ell_i$, $b_0 = 2|L_0|\,(\max(L_0) +3)$, $b_1 = 7^{|L_1|-1}(2\max(L_1)+1)$ and $L=\{\ell_1, \ell_2, \ldots, \ell_u\}$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.3: BuRi08
  • Theorem 2.4: BuDaTr22
  • Example 2.5
  • Theorem 2.6
  • proof
  • Theorem 3.1
  • ...and 4 more