A survey on constructive methods for the Oberwolfach problem and its variants

TL;DR

This survey reviews recent constructive methods for solving the Oberwolfach problem and its variants, focusing on graph decompositions into 2-regular subgraphs, with emphasis on blow-up constructions, bipartite factors, and circulant graphs.

Contribution

It compiles and analyzes recent advances in constructive techniques for the Oberwolfach problem, highlighting methods like blow-up constructions and automorphism-based solutions.

Findings

Progress in decomposing graphs into 2-regular factors.

Development of methods for bipartite and circulant graph solutions.

Enhanced understanding of automorphism-based constructions.

Abstract

The generalized Oberwolfach problem asks for a decomposition of a graph $G$ into specified 2-regular spanning subgraphs $F_1,\ldots, F_k$, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are pairwise isomorphic, and $G$ is the complete graph of odd order or the complete graph of even order with the edges of a $1$-factor removed. When there are two possible factor types, it is called the Hamilton-Waterloo problem. In this paper we present a survey of constructive methods which have allowed recent progress in this area. Specifically, we consider blow-up type constructions, particularly as applied to the case when each factor consists of cycles of the same length. We consider the case when the factors are all bipartite (and hence consist of even cycles) and a method for using circulant graphs to find solutions. We also consider constructions which yield solutions with well-behaved automorphisms.