A constructive solution to the Oberwolfach Problem with a large cycle

TL;DR

This paper presents a constructive method to solve the Oberwolfach problem for 2-regular graphs containing large cycles, advancing understanding of 2-factorizations of complete graphs.

Contribution

It introduces new constructions combining amalgamation-detachment with automorphism group techniques for large-cycle 2-regular graphs.

Findings

Constructs solutions for $OP(F)$ when $F$ has a cycle longer than a specific lower bound.

Uses amalgamation-detachment technique in combination with automorphism group methods.

Provides explicit conditions under which the Oberwolfach problem is solvable for large cycles.

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.