\documentclass{article}
\usepackage{fullpage,parskip}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{xcolor}
\usepackage{xfrac}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows,shapes,positioning}
\newcommand{\XXX}{{\color{red} \textsf{\textbf{XXX}}}}
\begin{document}
% To run this file, you'll need to have LaTeX installed. All of the
% lab computers have it. You can download a complete LaTeX
% distribution from
% https://www.tug.org/texlive/acquire-netinstall.html.
% Once you have LaTeX, you can either build your PDF on the
% command-line by running 'pdflatex hwXX.tex' to generate hwXX.pdf, or
% you can use an editor like LyX, TeXShop, or ShareLaTeX which
% automates the building of your PDF.
% Please print out your solution double-sided (a/k/a duplex) and bring
% it to class on the Wednesday it's due.
\noindent
% fill in the XXXs below
{\Large CS055 HW12 \qquad Name: \XXX \qquad CAS ID: \XXX} \\[.5em]
% I encourage you to collaborate, but please list any other students
% you talked to about the homework. If you worked alone, please just
% remove the XXXs.
Collaborators: \XXX
% Okay! Solve the problems below. please don't delete the problem
% statement or the ``enumerate'' bracketing which provides the
% numbering.
\begin{enumerate}
\item Draw an undirected graph with six or more nodes where every node
has a degree of at least 2.
% you'll want to leave this command in and then draw by hand.
% you can also use \includegraphics to include a PDF, if you like
% if you want to do something ``pro-level'', try using TikZ!
\vspace{2in}
\item The following questions all concern the number of possible edges
in a variety of different types of graphs with $n$ vertices. Read
closely!
\begin{enumerate}
\item What's the maximum number of edges in a directed graph with
$n$ vertices, if we disallow self-loops?
\textbf{Answer:} \XXX
\item What's the maximum number of edges in a directed graph with
$n$ vertices, if we allow self-loops?
\textbf{Answer:} \XXX
\item What's the maximum number of edges in an undirected graph with
$n$ vertices, if we disallow self loops?
\textbf{Answer:} \XXX
\item What's the maximum number of edges in an undirected graph with
$n$ vertices, if we allow self loops?
\textbf{Answer:} \XXX
\item Give a tight asymptotic bounds on the number of edges in a
graph with $n$ vertices. Use big-O notation.
\textbf{Answer:} \XXX
\end{enumerate}
\item Suppose $G$ is a simple (i.e., undirected) bipartite graph with
$n_1$ vertices in one partition and $n_2$ vertices in the other.
\begin{enumerate}
\item What's the maximum number of edges in $G$? Give a formula.
\textbf{Answer:} \XXX
\item Whats the minimum number of edges in $G$? Give a formula.
\textbf{Answer:} \XXX
\end{enumerate}
\item Determine whether $K_{2,2}$ (the complete bipartite graph on two
sets of two nodes each) is a subgraph of $K_4$ (the complete graph
on four nodes).
\textbf{Answer:} \XXX
\item Draw a strongly connected directed graph that has a simple circuit of
length 4 but no circuits of length 3 or less.
\vspace{2in}
\item Draw a tree with seven vertices. Draw the root at the top; there
should be three leaves at the bottom.
\vspace{2in}
\item For which numbers $n \ge 1$ is the complete graph $K_n$ on $n$
vertices a tree?
\textbf{Answer:} \XXX
\item For which numbers $m, n \ge 1$ is the complete bipartite graph
$K_{m,n}$ a tree? Explain (but don't prove) why. Don't write more
than five lines of text.
\textbf{Answer:} \XXX
\end{enumerate}
\end{document}