\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} %% header, footer definitions \def\course{MATH 314} \def\courselink{https://www.gsanmarco.com/graph-theory} \def\assignment{HW \#2} \def\assignlink{hw02.tex} \def\due{Due on Jan 31} \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} % 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}[1 pt] Recall that a \emph{directed graph} (or digraph) is a pair $D=(V,E)$ where $V$ is a set and $E\subseteq \{(u,v) \in V\times V \colon u\ne v\}$. For $u,v\in V$, a $u$-$v$ diwalk in $D$ is a sequence $(u=v_0,\dots,v_k=v)$ such that $(v_i,v_{i+1})\in E$ for all $i\in\{0,\dots,k-1\}$. Consider the relation $R$ on $V$ where $u\mathbin R v$ iff there is a $u$-$v$ diwalk. Give an example of a digraph $D$ where $R$ is \textbf{not} an equivalence relation. Justify your answer. (Feel free to draw a picture of $D$) \end{problem} \begin{problem}[2 pts] Let $G$ be a connected graph and consider a function $f\colon V(G)\to X$ where $X$ is some arbitrary set. Prove that if $f$ is \emph{not} a constant function, then there is an edge $uv\in E(G)$ such that $f(u)\neq f(v)$. \end{problem} \begin{problem}[2 pts] Prove that every graph on at least two vertices has a pair of vertices with the same degree. \end{problem} \begin{problem}[2 pts] Prove that if $G$ is a graph with $\delta(G)\geq 2$, then $G$ must contain a cycle. \end{problem} \begin{problem}[3 pts] Let $G$ be a graph and let $A$ be an independent set of $G$. Prove that \[ \sum_{v\in A}\deg v \leq\abs{E(G)} \] with equality if and only if $G$ is bipartite with parts $A$ and $V(G)\setminus A$. \noindent (Especially in this problem, be sure to carefully justify all steps in your argument) \noindent \textbf{Hint:} consider the sets $E_v=\{\text{edges incident with } v\}$ and $\widehat{E}=\bigcup_{v \in A} E_v$. For the equality, decide when is $\widehat{E}=E(G)$. \end{problem} \end{document}