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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 10 November 2017, 2:00 pm}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $G=G(A,B)$, with $|A|\leq |B|$. Define \[k = \max\{|S|-|N(S)|\ | \ S\subset A\}.\]
Show that a maximum matching in $G$ is of size $|A|-k$.

\item Show that every tree has at most one perfect matching.

\item Use Tutte's Theorem to show that every bridgeless cubic graph has a perfect matching.

\item Suppose you are running the Gale-Shapley algorithm on the university admittance problem, in which the students ask universities for entry, and the universities then select the students they most desire. In this case, however, each university $u_i$ has a capacity $c_i$ of students they can admit. Modify the Gale-Shapley algorithm for this situation, and show that your modification will produce a stable matching.
\end{enumerate}
\end{document}