% 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 paramter 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                   %
%%%%%%%%%%%%%%%%%%%%%
\newcommand{\myname}{Name}
\newcommand{\myandrew}{(andrewid)}
\newcommand{\mysection}{ - X}
\newcommand{\myhwnum}{7}
\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}{Given $k$ and a $k$-coloring of a $k$-chromatic graph, prove that for any color $c$ there is a vertex of color $c$ which is adjacent to vertices of every other color.}

\cleardoublepage

% Question 2
\question{2, Diestel 5.18}{Given a graph $G$ and $k \in \mathbb{N}$ let $P_G(k)$ denote the number of vertex colourings $V(G) \to \{1,\ldots,k\}$. Show that $P_G$ is a polynomial in $k$ of degree $n := |G|$, in which the coefficient of $k^n$ is 1 and the coefficient of $k^{n-1}$ is $-||G||$. ($P_G$ is called the \emph{chromatic polynomial} of $G$.)\\
(Hint. Apply induction on $||G||$.)}

\cleardoublepage

% Question 3
\question{3, Diestel 5.19}{Determine the class of all graphs $G$ for which $P_G(k) = k(k-1)^{n-1}$. (As in the previous exercise, let $n:= |G|$, and let $P_G$ denote the chromatic polynomial of $G$.)\\
Hint: A graph with $n$ vertices is a tree if and only if it is connected and has $n-1$ edges.}

\end{document}
