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Question

# At Rydell High School the senior class is represented on six school committees by Annemarie (A), Gary (G), Jill (J), Kenneth (K), Michael (M), Norma (N), Paul (P), and Rosemary (R). The senior members of these committees are {A, G, J, P}, {G, J, K, R},{A, M, N, P}, {A, G, M, N, P}, (A, G, K, N, R}, and {G, K, N, R}. (a) The student government calls a meeting that requires the presence of exactly one senior member from each committee. Find a selection that maximizes the number of seniors involved. (b) Before the meeting, the finances of each committee are to be reviewed by a senior who is not on that committee. Can this be accomplished so that six different seniors are involved in this review process? If so, how?

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DEFINITIONS

A $\textbf{matching}$ is a subset of the edges such that no two edges have a vertex in common.

A $\textbf{complete matching}$ from $V_1$ to $V_2$ means that every vertex from $V_1$ is the endpoint of an edge in the matching.

A $\textbf{bipartite graph}$ is a simple graph whose vertices can be partitioned into two sets $V_1$ and $V_2$ such that there are no edges among the vertices of $V_1$ and no edges among the vertices of $V_2$, while there can be edges between a vertex of $V_1$ and a vertex of $V_2$.

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