Lecture 010

Ada's Lecture

k-colorable: color vertex with k at most different colors so that no neighbors with the same color can be connected.

Bipartite: graph G = \langle{V, E}\rangle is bipartite iff it contains no cycles of odd length. (partitioned vertex into X, Y so that all edges have one endpoint in X and the other in Y.)

Matching: a subset of edges in bipartite graph where each vertex can only appear once.

Alternating (Augmenting) Path: a path alternate between M and not M

Exercise: proof a graph with all vertices degree at most 2 is either a path or a cycle

Algorithm

Algorithm

Hall's Theorem: Let G = \langle{X, Y, E}\rangle be a bipartite. There is a matching covering all the vertices in X (perfect matching) \iff (\forall S \subseteq X)(|S| \leq |N(S)|) (N is neighbors)

Hall's Theorem Application 1

Hall's Theorem Application 1

Hall's Theorem Application 2

Hall's Theorem Application 2

Application: randomly make 13 piles of 4 cards. show I can select a card from each pile to make up all 13 ranks

Stable Matching: // TODO: define

The Gale-Shapley proposal algorithm

Valid Partner: partners in at least one stable matching

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