Handshaking lemma

Handshaking lemma

In der Graphentheorie besagt das Handschlaglemma, dass in jedem Graph die Summe der Grade aller Knoten genau doppelt so groß ist wie die Anzahl seiner Kanten.

Formal heißt das: Ist G = (V,E) ein Graph (bei gerichteten Graphen werden sowohl die Ein- als auch die Ausgangs-Grade gezählt), so gilt also mit dem Grad d(v) eines Knotens v aus V:

\sum_{v\in V} d(v) = 2\cdot |E|

Daraus folgt sofort, dass jeder Graph eine gerade Anzahl von Knoten ungeraden Grades hat.

Bei regulären Graphen vereinfacht sich die Formel. Für einen k-regulären Graphen gilt:

k\cdot |V| = 2\cdot |E|

Literatur

  • Lutz Volkmann: Graphen und Digraphen, Springer (Wien) 1991, ISBN 3-211-82267-4, S.6

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