Perfect Matchings in a Planar Graphs

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Perfect Matchings in a Planar Graphs

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]

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