Graph Theory: Lecture No. 4
Tutte’s Condition:
(Let Gbe a graph, and let q(G)be the
number of odd components in G.)
For all S⊆V(G),q(G−S)≤ |S|.
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Graph Theory: Tutte's Condition and Perfect Matchings, Slides of Design Patterns
Tutte's condition in graph theory and how it relates to perfect matchings (1-factors) in a graph. That a graph satisfies tutte's condition if and only if it has a perfect matching. The document also discusses how to prove this converse and the concept of a 'bad set' in the context of tutte's condition.
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2012/2013
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Tutte’s Condition: (Let G be a graph, and let q(G ) be the number of odd components in G .) For all S ⊆ V (G ), q(G − S) ≤ |S|.
G has a perfect matching ( 1 -factor) if and only if Tutte’s condition is satisfied.
A set S, such that q(G − S) > |S| is a bad set. If G does not have a perfect matching, we have to demonstrate a bad set in G.
Instead of demonstrating a bad set in G , it is enough show a bad set in G + e, where e is a new edge. The same bad set set will be bad for G also. Thus we can choose to demonstrate a bad set in the edge maximal graph G ′^ produced from G by adding new edges, with respect to the property of not having a perfect matching.
We take S = {v ∈ V (G )|v is a universal vertex in G }. Our intention is to show that S has the special structure mentioned earlier.
Every bridge-less cubic graph has a perfect matching.