Petersen's theorem

A perfect matching (red edges), in the Petersen graph. Since the Petersen graph is cubic and bridgeless, it meets the conditions of Petersen's theorem.

A cubic (but not bridgeless) graph with no perfect matching, showing that the bridgeless condition in Petersen's theorem cannot be omitted

In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows:

Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching.

In other words, if a graph has exactly three edges at each vertex, and every edge belongs to a cycle, then it has a set of edges that touches every vertex exactly once.

Proof

We show that for every cubic, bridgeless graph Template:Math we have that for every set Template:Math the number of connected components in the graph induced by Template:Math with an odd number of vertices is at most the cardinality of Template:Math. Then by the Tutte theorem Template:Math contains a perfect matching.

Let Template:Math be a component with an odd number of vertices in the graph induced by the vertex set Template:Math. Let Template:Math denote the vertices of Template:Math and let Template:Math denote the number of edges of Template:Math with one vertex in Template:Math and one vertex in Template:Math. By a simple double counting argument we have that

$\sum _{v\in V_i}{\mathrm{deg}}_G(v)=2|E_i|+m_i,$

where Template:Math is the set of edges of Template:Math with both vertices in Template:Math. Since

$\sum _{v\in V_i}{\mathrm{deg}}_G(v)=3|V_i|$

is an odd number and Template:Math is an even number it follows that Template:Math has to be an odd number. Moreover, since Template:Math is bridgeless we have that Template:Math.

Let Template:Math be the number of edges in Template:Math with one vertex in Template:Math and one vertex in the graph induced by Template:Math. Every component with an odd number of vertices contributes at least 3 edges to Template:Math, and these are unique, therefore, the number of such components is at most Template:Math. In the worst case, Template:Math is an independent set, and therefore Template:Math. We get

$|U|\ge \frac{1}{3}m\ge |\{\text{components with an odd number of vertices}\}|,$

which shows that the condition of Tutte theorem holds.

History

The theorem is due to Julius Petersen, a Danish mathematician. It can be considered as one of the first results in graph theory. The theorem appears first in the 1891 article "Die Theorie der regulären graphs". By today's standards Petersen's proof of the theorem is complicated. A series of simplifications of the proof culminated in the proofs by Template:Harvtxt and Template:Harvtxt.

In modern textbooks Petersen's theorem is covered as an application of Tutte's theorem.

Applications

• In a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length three.
• Petersen's theorem can also be applied to show that every maximal planar graph can be decomposed into a set of edge-disjoint paths of length three. In this case, the dual graph is cubic and bridgeless, so by Petersen's theorem it has a matching, which corresponds in the original graph to a pairing of adjacent triangle faces. Each pair of triangles gives a path of length three that includes the edge connecting the triangles together with two of the four remaining triangle edges.
• By applying Petersen's theorem to the dual graph of a triangle mesh and connecting pairs of triangles that are not matched, one can decompose the mesh into cyclic strips of triangles. With some further transformations it can be turned into a single strip, and hence gives a method for transforming a triangle mesh such that its dual graph becomes hamiltonian.Template:Sfnp

Extensions

Each edge belongs to some Perfect matching in cubic bridgeless graphs

Schönberger strengthened Petersen's theorem in 1934 by showing that each edge of any cubic bridgeless graph belongs to some perfect matching.

Number of perfect matchings in cubic bridgeless graphs

It was conjectured by Lovász and Plummer that the number of perfect matchings contained in a cubic, bridgeless graph is exponential in the number of the vertices of the graph Template:Math.Template:Sfnp The conjecture was first proven for bipartite, cubic, bridgeless graphs by Template:Harvtxt, later for planar, cubic, bridgeless graphs by Template:Harvtxt. The general case was settled by Template:Harvtxt, where it was shown that every cubic, bridgeless graph contains at least $2^{n/3656}$ perfect matchings.

Algorithmic versions

Template:Harvtxt discuss efficient versions of Petersen's theorem. Based on Frink's proof they obtain an Template:Math algorithm for computing a perfect matching in a cubic, bridgeless graph with Template:Math vertices. If the graph is furthermore planar the same paper gives an Template:Math algorithm. Their Template:Math time bound can be improved based on subsequent improvements to the time for maintaining the set of bridges in a dynamic graph.Template:Sfnp Further improvements, reducing the time bound to Template:Math or (with additional randomized data structures) Template:Math, were given by Template:Harvtxt.

Higher degree

If G is a regular graph of degree d whose edge connectivity is at least d − 1, and G has an even number of vertices, then it has a perfect matching. More strongly, every edge of G belongs to at least one perfect matching. The condition on the number of vertices can be omitted from this result when the degree is odd, because in that case (by the handshaking lemma) the number of vertices is always even.

See also

• 2-factor theorem – related theorem by Petersen

References

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