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b/übung_5/aufgabe_2.tex @@ -1,8 +1,17 @@ \section{Hamiltonkreise} \begin{tasks} \item + Ein Kreis hat im allgemeinen mindestens die Länge 3. Da der Turniergraph mindestens + 3 Knoten enthält und stark zusammenhängend ist, liegt jeder Knoten auf einem Kreis + mit Länge $\geq 3$. \points{1} \item + Gegeben ist ein Kreis $K$ mit Länge $\geq 3$. + Um $K$ zu verkleinern wählen wir uns drei aufeinanderfolgende Knoten $a, b, c$ aus. + Nach Definition gibt es für jedes Knotenpaar eine gerichtete Kante. Wir ersetzen + jetzt einen Weg der Länge 2 durch eine Kante die nach Annahme existieren muss. Das + machen wir solange, bis der Kreis die Länge 3 hat. + \points{2} \item \points{1}