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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Matchings with Tuttes Theorem Last week we saw a fairly strong necessary criterion for a graph to have a perfect matching. Today we see that this condition is in fact sufficient. Theorem 1 (Tutte, 47) . If there is no set S V ( S ) such that G- S has more than | S | components with an odd number of vertices, then G has a perfect matching. Proof. Let us refer to the condition given above that there is no set S V ( S ) such that G- S has more than | S | components with an odd number of vertices, for brevity, as Tuttes criterion. We already showed previously that any graph not satisfying Tuttes criterion cannot have a perfect matching; we thus now want to show that all graphs satisfying Tuttes criterion do have a perfect matching. One method we can use to further specify the set of graphs over which we have to prove this is to observe the effect of adding an edge to a graph which meets Tuttes Criterion results in a graph which still meets Tuttes Criterion. Suppose G satisfies Tuttes criterion; it is fairly easy to show that G = G + e satisfies it as well. Consider a set S of vertices from the shared vertex-set of G and G . We know that G- S has no more than | S | odd components. G- S is either identical to G- S (if e is incident on a vertex of S ) or identical to G- S + e (otherwise). In the first instance, G- S clearly has the same number of odd components as G- S . In the second, either e is internal to a component of G- S , in which case G- S has the same number of odd components as G- S , or it is between two components of G- S , in which case the number of components in G- S may be different, but it may be easily seen that G- S has no more odd components than G- S : an edge bridging two even components, or connecting an odd component to an even component, has no effect on the number of odd components, while an edge connecting two odd components (and thereby joining them into an even component) reduces the number of odd components by two. Thus, the number of odd components in G- S is at most equal to the number of odd components in G , so if G satisfies the Tutte criterion, so does G . It is even easier, of course, to show that if G has a perfect matching, so does G . As a practical upshot of this, we can see that if G satisfies both the condition and consequence of Tuttes theorem, so does any graph resulting from adding edges to G , so we only need demonstrate Tuttes theorem for edge-minimal graphs which satisfy Tuttes criterion. We may thus inspect a very specific set of graphs. Starting from a graph consisting only of an even number of isolated vertices (if | G | is odd, it clearly does not satisfy Tuttes criterion, and is not of interest here) and adding edges sequentially, we are guaranteed two phases, and we posit the possibility of a third: when there are very few edges, both Tuttes criterion and existence of a perfect matching are false. This sequence terminates in a complete grapha perfect matching are false....
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