\documentclass[11pt]{scrartcl}
\usepackage[headheight=1pt, includeheadfoot, headsep=1cm, top=1.5cm, bottom=1.3cm, left=2cm, right=2cm]{geometry}
\usepackage{mathtools}
\usepackage{amsmath,amsthm, amscd, amssymb, amsfonts}
\usepackage[english]{babel}
\usepackage{fancyhdr}
\usepackage{enumitem}
\setlist{itemsep=0.5em}
\usepackage{xcolor}
\definecolor{forestgreen}{rgb}{0.13, 0.55, 0.13}
\usepackage[colorlinks = true,
linkcolor = forestgreen,
urlcolor  = forestgreen,
citecolor = forestgreen,
anchorcolor = forestgreen]{hyperref}
\usepackage{multicol}
\usepackage{multienum}
\usepackage{euler}
\usepackage[OT1]{eulervm}
\renewcommand{\rmdefault}{pplx}
\usepackage{tikz}
\usepackage{mathabx}
\usepackage{lastpage}



\usepackage{tikz}
\usepackage{verbatim}
\usetikzlibrary{shapes.geometric,shapes.multipart,arrows,calc}


\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}			
\tikzset{
	edge/.style = {black, ultra thick, outer sep=2pt},
	vertex/.style = {draw=black, very thick, fill=white, circle, minimum width=8pt, inner sep=2pt, outer sep=1pt},
	arc/.style = {black, ultra thick, outer sep=2pt, decoration={markings, mark=at position #1 with {\arrow[scale=1.5]{latex}}}, postaction=decorate},
	arc/.default = 0.58
}


%% header, footer definitions
\def\course{MATH 314}
\def\courselink{https://www.gsanmarco.com/graph-theory}
\def\assignment{HW \#7}
\def\assignlink{hw07.tex}
\def\due{Due on Mar 21}




\pagestyle{fancy}
\fancyhead[L]{\course}
\fancyhead[C]{\assignment} 
\fancyhead[R]{\due} 
%\renewcommand{\footrulewidth}{0.4pt}
%\fancyfoot{}
%\fancyfoot[R]{\fontsize{8}{10} \selectfont  Page \thepage~of \pageref{LastPage}}
%\fancyfoot[L]{\fontsize{10}{12} \selectfont  Source file available at \href{\courselink}{\textbf{\courselink}}}

%% environments
% theorem style
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{defn}[theorem]{Definition}

% definition style
\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem{Solution}{Solution}
% custom
\newenvironment{solution}{\noindent\textbf{Solution.}}{\qed}

%% custom commands

\newcommand*{\abs}[1]{\lvert #1\rvert}     % absolute values, cardinality
\newcommand*{\eps}{\varepsilon}            % prettier epsilon
\newcommand*{\R}{\mathbb{R}}               % real numbers
\newcommand*{\C}{\mathbb{C}}               % complex numbers
\newcommand*{\Q}{\mathbb{Q}}               % rational numbers
\newcommand*{\Z}{\mathbb{Z}}               % integers
\newcommand*{\N}{\mathbb{N}}               % natural numbers
\newcommand*{\F}{\mathbb{F}}               % field
\newcommand*{\family}{\mathcal{F}}         % family
\newcommand*{\GL}{\mathbf{GL}}             % general linear group
\newcommand*{\what}[1]{\widehat{#1}}       % wider hat
\newcommand*{\wbar}[1]{\overline{#1}}      % wider bar
\newcommand*{\mcal}[1]{\mathcal{#1}}       % mathcal
\newcommand*{\mbf}[1]{\mathbf{#1}}         % mathbf
\newcommand*{\mbb}[1]{\mathbb{#1}}         % mathbb

% operators
\let\Pr\relax\DeclareMathOperator*{\Pr}{\mathbf{Pr}}       % probability
\DeclareMathOperator*{\E}{\mathbb{E}}          % expectation
\DeclareMathOperator*{\Var}{\mathbf{Var}}      % variance
\DeclareMathOperator*{\Span}{span}             % span
\DeclareMathOperator{\tr}{tr}                  % trace
\DeclareMathOperator{\rank}{rank}              % rank
\DeclareMathOperator{\supp}{supp}              % support
\DeclareMathOperator{\diam}{diam}



\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}[2pts]
	For infinitely many integers $n$, construct a graph $G$ with the following properties:
	\begin{enumerate}[label=$\bullet$]
		\item $G$ is connected and has $n$ vertices, and
		\item $G$ is \emph{not} a tree, and
		\item Every $v\in V(G)$ which is not a leaf of $G$ is a cut-vertex of $G$.
	\end{enumerate}
\end{problem}


\begin{problem}[1 + 1 pts]\label{spanning}
	Let $G$ be a graph and suppose that $H$ is any spanning subgraph of $G$.
	\begin{enumerate}[label=(\alph*)]
		\item Prove that $\lambda(G)\geq\lambda(H)$.
		\item Prove that $\kappa(G)\geq\kappa(H)$.
	\end{enumerate}
\end{problem}



\begin{problem}[1 + 2 pts]\label{delete_edge}
	Let $G=(V,E)$ be a graph with at least one edge and fix any $e\in E$.
	\begin{enumerate}[label=(\alph*)]
		\item Prove that $\lambda(G)\geq\lambda(G-e)\geq\lambda(G)-1$.
		\item Prove that $\kappa(G)\geq\kappa(G-e)\geq\kappa(G)-1$.
	\end{enumerate}
\end{problem}


\begin{problem}[1 pts]
	For each non-negative integer $k$, find an example of a graph $G$ with $\kappa(G)=\lambda(G)=1$, yet there is some vertex $v\in V(G)$ such that $\kappa(G-v)=\lambda(G-v)=k$.
	
	That is to say, the natural analogue of Problem~\ref{delete_edge} fails when deleting vertices instead of edges.
\end{problem}


\begin{problem}[2 pts]
	Let $G$ be a graph.
	The \emph{$k$-th power of $G$} is the graph $G^k$ which has the same vertex-set as $G$ and $uv\in E(G^k)$ iff $d_G(u,v)\leq k$.
	
	Prove that if $G$ is a connected graph on at least $k+1$ vertices, then $G^k$ is $k$-connected.
	
\end{problem}


\end{document}