2-factor theorem

In the mathematical discipline of graph theory, 2-factor theorem discovered by Julius Petersen, is one of the earliest works in graph theory and can be stated as follows:

2-factor theorem. Let G be a regular graph whose degree is an even number, 2k. Then the edges of G can be partitioned into k edge-disjoint 2-factors.

Here, a 2-factor is a subgraph of G in which all vertices have degree two; that is, it is a collection of cycles that together touch each vertex exactly once.

Proof

In order to prove this generalized form of the theorem, Petersen first proved that a 4-regular graph can be factorized into two 2-factors by taking alternate edges in a Eulerian trail. He noted that the same technique used for the 4-regular graph yields a factorization of a 2k-regular graph into two k-factors.

To prove this theorem, first we remark that it is sufficient to consider connected graphs. A connected graph with even degree has an Eulerian trail. Traversing this Eulerian trail we get an orientation D of G such that every point has indegree and outdegree = k. Then we replace every point v ϵ V(D) by two points v’, v”, and for every directed line uv ϵ E(D) we draw in one line from u’ to v”. Since D has in- and outdegree equal to k the resulting bigraph G’ is k-regular. The lines of G’ can be decomposed into k perfect matchings. Now if we identify v’ and v” for every v, we recover the graph G, and these k perfect matchings of G’ will be mapped onto k 2-factors of G which partition the lines.^[1]

In other words, for the general case the factorization of a 2k-regular graph into two factors is an easy consequence of Euler’s theorem: By applying the same argument to each component, we may assume that G is connected and 2k-regular with vertices v₁, ..., v_n. Let C be an Eulerian circuit of G, followed in a particular direction. Then use Eulerian circuit of G to create an supplementary k-regular bipartite graph H, such that a perfect matching in H corresponds to a 2-factor in G.

History

The theorem was discovered by Julius Petersen, a Danish mathematician. It is in fact, one of the first results in graph theory. The theorem appears first in the 1891 article "Die Theorie der regulären graphs". To prove the theorem Petersen's fundamental idea was to 'colour' the edges of a trial or a path alternatingly red and blue, and then to use the edges of one or both colours for the construction of other paths or trials.