By Berge C.
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Extra resources for Hypergraphs
Clearly, each matching in our sequence is better than the previous, and it is easy to check inductively that they all keep the vertices in A happy. So the sequence continues until it terminates with a matching M such that every unmatched vertex in A is inacceptable to all its neighbours in B. As every matched vertex in A is happy with M , this matching is stable. 1 Matching in bipartite graphs Despite its seemingly narrow formulation, the marriage theorem counts among the most frequently applied graph theorems, both outside graph theory and within.
Or for given k: which graphs can be partitioned into at most k forests? An obvious necessary condition now is that every set U ⊆ V (G) induces at most k (|U | − 1) edges, no more than |U | − 1 for each forest. 4 Tree-packing and arboricity 49 Once more, this condition turns out to be suﬃcient too. 4. (Nash-Williams 1964) A multigraph G = (V, E) can be partitioned into at most k forests if and only if G [ U ] k (|U | − 1) for every non-empty set U ⊆ V . Proof . The forward implication was shown above.
A typical situation is that vertices are not indiﬀerent to which of their incident edges are picked to match them, but prefer some to others. Then if M is a matching and e = ab is an edge not in M such that both a and b prefer e to their current matching edge (if they are matched), then a and b may agree to change M locally by including e and discarding their earlier matching edges. The matching M , although perhaps of maximal size, would thus be unstable. More formally, call a family ( v )v ∈ V of linear orderings v on E(v) a set of preferences for G.