A constructive solution to the Oberwolfach Problem with a large cycle

Tommaso Traetta

doi:10.1016/j.disc.2024.113947

paper

A constructive solution to the Oberwolfach Problem with a large cycle

arXiv:2306.12713 · doi:10.1016/j.disc.2024.113947

Abstract

For every $2$ -regular graph $F$ of order $v$ , the Oberwolfach problem $O P (F)$ 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 $O P (F)$ 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.

14 pages