Perfect Matchings in a Planar Graphs

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Perfect Matchings in a Planar Graphs

Perfect Matchings in a Planar Graphs

Recall the definitions of:

-matchings

- perfect matchings

-planar graphs

Recall the (monomer-)dimer problem from physics.

[Section 1.2]


Perfect Matchings in a Planar Graphs

Fact 1.9:

Let M, M’ be two perfect matchings, then M ∪ M’ forms a

collection of single edges and even-length cycles.

[Section 1.2]


Perfect Matchings in a Planar Graphs

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 matchings 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 Matchings in a Planar Graphs

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 Matchings in a Planar Graphs

Thm 1.11 (Kasteleyn):

For any Pfaffian orientation G → ,

# perfect matchings 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 deciding if a

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


Proving Kasteleyn’s Theorem

[Section 1.2]

Def: Let G ↔ be obtained from G by replacing 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 matchings and even cycle covers of G ↔ .


Proving Kasteleyn’s Theorem

Thm 1.11: If G → Pfaffian, then

Proof:

(# perfect matchings 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 having 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 inside 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 having 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 inside C.

Proof:

Let v = # vertices inside C

k = length of C

c = # clockwise edges in C

f = # faces inside C

e = # edges inside C

c i = # clockwise edges in component i inside C

[Section 1.2]


Pfaffian orientations in planar graphs

Thm 1.14: Every planar graph has a Pfaffian orientation.

Proof:

[Section 1.2]

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