A 2-factor of a graph is a spanning subgraph in which every vertex has vertex degree 2. Thus a 2-factor of a finite simple graph is a disjoint union of cycle graphs covering all vertices. A Hamiltonian cycle is a connected 2-factor.
Every regular graph of positive even vertex degree has a 2-factor and admits a decomposition into edge-disjoint 2-factors.
Lee (2026) gives an asymptotically sharp counting
version of this theorem. For an 
-vertex 
-regular graph 
, let 
be its set of 2-factors. Then
 |
where 
as 
, uniformly over simple
graphs 
.
The asymptotic formula is therefore a growing-degree
statement, not a fixed-degree estimate as 
.
Lee (2026) reports that ChatGPT helped develop an auxiliary proof under his guidelines, while he supplied the high-level ideas and verified the results.
