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\begin{document}
\title{Math 301: Homework 6}
\author{Mary Radcliffe}
\date{due 2 Nov 2016}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $G$ be a graph of order $n\geq 4$ and having $m\geq \frac{n}{2}\sqrt{n-1}+1$ edges. Prove that $G$ contains either a 3-cycle or a 4-cycle.

\item \begin{enumerate}
\item Let $G=(V, E)$ be a graph with average degree $D$. Show that there exists a subgraph $H$ in $G$ with $\delta(H)\geq \frac{D}{2}$, that is, the minimum degree in $H$ is at least $\frac{D}{2}$.
\item Let $T$ be a tree on $k$ vertices. Prove that $\frac{n(k-2)}{2}\leq ex(n, T)\leq nk$ for any $n\in \N$ having $n$ divisible by $k-1$. 
\end{enumerate}

Note: We do not know $ex(n, T)$ for an arbitrary $T$. It is conjectured that the lower bound above is tight, but it is not proven. It is certainly tight for some trees (paths, stars, probably others).


\end{enumerate}

\end{document}