A complete solution to the infinite Oberwolfach problem

TL;DR

This paper provides a comprehensive solution to the infinite Oberwolfach problem, establishing conditions for the existence of 2-factorizations in infinite complete graphs with specific symmetry properties.

Contribution

It introduces a complete solution to the infinite Oberwolfach problem, including cases with group actions and locally finite subgraphs, expanding the understanding of infinite graph factorizations.

Findings

Proves existence of regular solutions under involution-free group actions.

Provides solutions for locally finite subgraphs of infinite complete graphs.

Characterizes subgraphs that can be included in solutions to the infinite Oberwolfach problem.

Abstract

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we characterize the infinite subgraphs $L$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(L)$.