CS 594 Graph Theory, Spring 2014 Homework 3
1. For which values of m and n does Km,n contain a perfect matching? For those values, how
many perfect matchings are there?
2. Prove or disprove:
a. If v is a cut vertex in G, then G contains a maximum matching which does not include v.
b. If v is a cut vertex in G, then G contains a maximal matching which does not include v.
3. What is the maximum number of perfect matchings in an even cycle with n chords? Explain.
4. How many maximal matchings are in Kn?
5. Consider a two-player game in which player 1 chooses a vertex v in a graph G, player 2 selects
a vertex u adjacent to v, player 1 selects a vertex w adjacent to u, and so on, each player choosing
a previously unchosen vertex. The game continues until one player cannot select a vertex, at
which point the other player wins. Show that if G has a perfect matching, player 2 can always
win, and if G does not have a perfect matching, player 1 can always win.
6. Prove that a tree T has a perfect matching if and only if o(T – v) = 1 for every v‫ ג‬T.