% To print single-sided: delete ",twoside" below
\documentclass[11pt,twoside]{article}

% Some very basic packages. Feel free to add your favorites!
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{fancyhdr}
\usepackage{graphicx}

% Set up the page correctly
\addtolength{\oddsidemargin}{-.54in}
\addtolength{\evensidemargin}{-1.2in}
\addtolength{\textwidth}{1.75in}
\addtolength{\topmargin}{-.875in}
\addtolength{\textheight}{1.75in}
\setlength{\parindent}{0pt}
\setlength{\parskip}{5pt plus 1pt}

% Some nifty commands. Thanks Adam Blank!
\newcommand{\question}[2] {\vspace{.25in} \hrule\vspace{0.5em}
\noindent{{\bf #1:} #2} \vspace{0.5em}
\hrule \vspace{.10in}}
\renewcommand{\part}[1] {\vspace{.10in} {\bf (#1)}}

% Add an image centered on the page.
% The first parameter is a scale factor (set to 1 for no scale)
% The second parameter is the image file. Make sure it is in the same folder as your .tex file!
\newcommand{\centergfx}[2]{\begin{center}\includegraphics[scale = #1]{#2}\end{center}}

% Command to insert a minus sign with a dot over it. Stolen from
% http://www.cs.miami.edu/~burt/learning/Math688.931/Other/errata.tex
\newcommand{\monus}{\stackrel{{}^{\scriptstyle .}}{\smash{-}}}

%%%%%%%%%%%%%%%%%%%%%
%		Replace "Name" and                 %
%		   "andrewid" below                   %
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\newcommand{\myname}{Name}
\newcommand{\myandrew}{(andrewid)}
\newcommand{\mysection}{ - X}
\newcommand{\myhwnum}{4}
\newcommand{\myclass}{21-484 Graph Theory}

% Set up the nice header
\pagestyle{fancyplain}
\lhead{\fancyplain{}{\textbf{HW\myhwnum}}}
\rhead{\fancyplain{}{\myname \ \myandrew \ \mysection}}
\chead{\fancyplain{}{\myclass}}

\begin{document}
% Question 1
\question{1, Diestel 3.4}{Let $X$ and $X'$ be minimal separators in $G$ such that $X$ meets (intersects non-trivially) at least
two components of $G - X'$. Show that $X'$ meets all the components of $G - X$, and that $X$ meets all the
components of $G - X'$.}

\cleardoublepage

% Question 2
\question{2, Diestel 3.7}{Show the block graph of any connected graph is a tree.}

\cleardoublepage

% Question 3
\question{3, Diestel 3.8}{Let $G$ be a $k$-connected graph, and let $xy$ be an
edge of $G$. Show that $G / xy$ is $k$-connected if and only if $G-\{x,y\}$ is $(k-1)$-connected.}

\cleardoublepage

% Question 4
\question{4, Diestel 3.9}{(i) Let $e$ be an edge in a 2-connected graph $G \neq K^3$. Show that either $G - e$ or $G/e$ is again 2-connected.
\newline
\vspace{.1truein}

(ii) Does every 2-connected graph $G \neq K^3$ have an edge $e$ such that $G/e$ is still 2-connected?}

\cleardoublepage

% Question 5
\question{5, Diestel 3.14}{Show that every transitive graph $G$ with $\kappa(G) = 2$ is a cycle. Hint: Exercise 3.4 is useful.}


\end{document}
