Perfect Matchings in a Planar Graphs

**Perfect** **Match ings**

Recall the def**in**itions of:

-match**in**gs

- perfect match**in**gs

-planar graphs

Recall the (monomer-)dimer problem from physics.

[Section 1.2]

**Perfect** **Match ings**

Fact 1.9:

Let M, M’ be two perfect match**in**gs, then M ∪ M’ forms a

collection of s**in**gle edges and even-length cycles.

[Section 1.2]

**Perfect** **Match ings**

Def:

Let G → be an orientation of G and let C be a cycle of G.

C is oddly oriented if odd # edges **in** C disagrees with G → .

G → is Pfaffian if, for every perfect match**in**gs M,M’, every

cycle **in** M ∪ M’ is oddly oriented.

Note: not every cycle **in** G, just every cycle **in** M ∪ M’

[Section 1.2]

**Perfect** **Match ings**

Def:

Skew adjacency matrix of G → is A s (G → ) where

+1 if (i,j) is an edge **in** G →

a ij = -1 if (j,i) is an edge **in** G →

0 otherwise

[Section 1.2]

**Perfect** **Match ings**

Thm 1.11 (Kasteleyn):

For any Pfaffian orientation G → ,

# perfect match**in**gs of G = det A s (G → )

Thm 1.14: Every planar graph has a Pfaffian orientation

(and it can be easily constructed).

[Section 1.2]

Note: It is not known whether the problem of decid**in**g if a

graph has a Pfaffian orientation is **in** P nor whether it is NPcomplete.

Prov**in**g Kasteleyn’s Theorem

[Section 1.2]

Def: Let G ↔ be obta**in**ed from G by replac**in**g every edge

by . An even cycle cover of G ↔ is a collection of even

cycles that cover every vertex exactly once.

Lemma 1.12: There is a bijection between (ordered) pairs of

perfect match**in**gs and even cycle covers of G ↔ .

Prov**in**g Kasteleyn’s Theorem

Thm 1.11: If G → Pfaffian, then

Proof:

(# perfect match**in**gs of G) 2 = det A s (G → )

[Section 1.2]

Pfaffian orientations **in** planar graphs

Lemma 1.13: Let G → be a connected planar digraph (drawn **in**

plane) with all faces except outer hav**in**g an odd number of

clockwise oriented edges. Then, **in** any (simple) cycle C, the

number of clockwise edges is opposite parity as the number of

vertices **in**side C.

In particular, G → from Lemma 1.13 is Pfaffian.

[Section 1.2]

Pfaffian orientations **in** planar graphs

Lemma 1.13: Let G → be a connected planar digraph (drawn **in**

plane) with all faces except outer hav**in**g an odd number of

clockwise oriented edges. Then, **in** any (simple) cycle C, the

number of clockwise edges is opposite parity as the number of

vertices **in**side C.

Proof:

Let v = # vertices **in**side C

k = length of C

c = # clockwise edges **in** C

f = # faces **in**side C

e = # edges **in**side C

c i = # clockwise edges **in** component i **in**side C

[Section 1.2]

Pfaffian orientations **in** planar graphs

Thm 1.14: Every planar graph has a Pfaffian orientation.

Proof:

[Section 1.2]