\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsfonts,amsmath,amssymb,amsthm}

\PassOptionsToPackage{obeyspaces}{url}
\usepackage[colorlinks=true,citecolor=blue,urlcolor=blue,linkcolor=blue,bookmarksopen=true]{hyperref}
\usepackage{tikz}

\usepackage{enumitem}
\setlist{itemsep=0.5em}

%% header definitions
\def\course{MATH 314}
\def\assign{HW \#7}
\def\due{Mar 22}
\def\doclink{\url{https://mathematicaster.org/teaching/graphs2022/hw/hw7.tex}}

%% 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}

% custom
\newenvironment{solution}{\paragraph{Solution.}}{\qed}

%% custom commands
\newcommand*{\arXiv}[1]{\href{http://arxiv.org/pdf/#1}{arXiv:#1}}       % arXiv link
\newcommand*{\eqdef}{\stackrel{\mbox{\normalfont\tiny{def}}}{=}}        % definition by equality
\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}
\DeclareMathOperator{\Aut}{Aut}



\begin{document}
\pagestyle{empty}

\noindent
\begin{minipage}{.33\textwidth}
	\flushleft {\bf \course}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\centering {\bf \assign}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\flushright {\bf \due}
\end{minipage}
\vskip3pt
\hrule\hrule\hrule
\vskip5pt

\noindent {\footnotesize This document is from \doclink}
\vskip10pt

\noindent {\small 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.
}

\vskip20pt

\begin{problem}[2pts]
    For infinitely many integers $n$, construct a graph $G$ with the following properties:
    \begin{itemize}
        \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{itemize}
    ``For infinitely many integers $n$...'' means the following: Pick your favorite infinite subset $A\subseteq\N$ and, for each $n\in A$, build the desired graph for that $n$.
    For instance, maybe you pick $A$ to be the set of even naturals, or maybe you pick $A$ to be the set of integers of the form $2^k$, or maybe you pick $A$ to be the set of all primes larger than $100^{100^{100^{100}}}$, etc.
    As long as $A$ is infinite, you're fine.
\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}
        \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}
        \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.

    (You don't have to turn this is, but you should convince yourself that it's true: $G^k$ is a clique if and only if $\diam(G)\leq k$.)
\end{problem}





\end{document}
