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%% header, footer definitions
\def\course{MATH 314}
\def\courselink{https://www.gsanmarco.com/graph-theory}
\def\assignment{HW \#3}
\def\assignlink{hw03.tex}
\def\due{Due on Feb 7}




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\begin{document}
\noindent  Unless explicitly requested by a problem, do not include sketches as part of your proof. You are free to use the result from any problem on this (or previous) assignment as a part of your solution to a different problem even if you have not solved the former problem.
\vskip10pt


\begin{problem}[1 pt]
Show that if $G$ is a connected graph that is not regular, then $G$ contains \textbf{adjacent} vertices $u$ and $v$ such that $\deg u \ne \deg v$.

\textbf{Hint}: there is a reason why this problem is worth 1 pt.
\end{problem}

\begin{problem}[2 pts]
	Let $G$ be a graph on $n$ vertices.
	Prove that if $\Delta(G)+\delta(G)\geq n-1$ then $G$ is connected.
\end{problem}



\begin{problem}[2 pts]
	Let $G$ be a graph with the property that $\deg u+\deg v\equiv 1\pmod 2$ for every $uv\in E(G)$.
	Prove that $\abs{E(G)}$ is even.
\end{problem}


\begin{problem}[2 pts]
	Let $G$ be a connected graph and suppose that $f\colon V(G)\to\Z$ is a function with the property that $f(u)+f(v)\equiv 0\pmod 3$ for every $uv\in E(G)$.
	Prove that if $G$ is \emph{not} bipartite, then $f(v)\equiv 0\pmod 3$ for every $v\in V(G)$.
\end{problem}


\begin{problem}[2 pts]
	Let $G$ be a graph on $n$ vertices.
	Prove the following:
	\begin{enumerate}
		\item $\delta(G)+\delta(\wbar G)\leq n-1$,
		 \item $\delta(G)+\delta(\wbar G)= n-1$ if and only if $G$ is regular.
	\end{enumerate}
\end{problem}


\begin{problem}[0.5+0.5 pts]
	Determine whether or not the following sequences are graphical.
	If the sequence is graphical, draw a picture of a graph with that degree sequences.
	If the sequence is not graphical, prove this.
	\begin{enumerate}
		\item $5,3,3,3,3,2,2,2,1$
		\item $6,5,5,4,3,2,1$
	\end{enumerate}
\end{problem}




\end{document}