\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 \#5} \def\assignlink{hw05.tex} \def\due{Due on Feb 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}[2 pts] Fix an integer $n\geq 2$. Prove that a sequence of integers $d_1,\dots,d_n$ is the degree sequence of some tree if and only if $d_i\geq 1$ for all $i\in\{1, \dots, n\}$ and $\displaystyle \sum_{i=1}^n d_i=2n-2$. \end{problem} \begin{problem}[2 pts] This problem gives an alternative proof for the existence of spanning trees. Let $G$ be a connected graph. Consider the family $\mathcal F$ of all spanning subgraphs of $G$ that contain no cycle. Show that $\mathcal F$ is non empty and that an element of $\mathcal F$ with \textbf{maximum} number of edges is a spanning tree of $G$. \end{problem} \begin{problem}[2 pts] Let $G$ be a connected graph and let $S$ be any subset of edges of $G$. Show that $G$ has a connected, spanning subgraph $H$ such that $S\subseteq E(H)$ and if $C$ is a cycle in $H$, then $E(C) \subseteq S$. \end{problem} \begin{problem}[2 pts] Let $G$ be a connected graph and let $w\colon E(G)\to\mathbb R^{>0}$ be a weight function. Show that if all weights are distinct (that is $w(e)\neq w(s)$ for all distinct $e,s\in E(G)$), then $G$ has a \emph{unique} minimum spanning tree. \end{problem} \begin{problem}[1+ 1 pts] Consider the following weighted graph \begin{center} \begin{tikzpicture}[scale=1] \useasboundingbox (-4.5,-2) rectangle (2.5,2); \begin{pgfonlayer}{foreground} \node[vertex] (a) at (-4,0) {}; \node[vertex] (b) at (-2,2) {}; \node[vertex] (c) at (-2,-2) {}; \node[vertex] (d) at (0,0) {}; \node[vertex] (e) at (2,2) {}; \node[vertex] (f) at (2,-2) {}; \node[left,xshift=-3pt] at (a) {\large $a$}; \node[right,xshift=3pt] at (b) {\large $b$}; \node[right,xshift=3pt] at (c) {\large $c$}; \node[right,xshift=3pt] at (d) {\large $d$}; \node[right,xshift=3pt] at (e) {\large $e$}; \node[right,xshift=3pt] at (f) {\large $f$}; \end{pgfonlayer} \begin{pgfonlayer}{background} \draw[edge] (a) -- (b) node[midway,left,yshift=2pt] {\large $1$}; \draw[edge] (a) -- (c) node[midway,left,yshift=-2pt] {\large $2$}; \draw[edge] (b) -- (c) node[midway,left,xshift=2pt] {\large $4$}; \draw[edge] (b) -- (d) node[midway,right,yshift=2pt] {\large $2$}; \draw[edge] (c) -- (d) node[midway,right,yshift=-2pt] {\large $3$}; \draw[edge] (d) -- (e) node[midway,left,yshift=2pt] {\large $4$}; \draw[edge] (d) -- (f) node[midway,left,yshift=-2pt] {\large $5$}; \draw[edge] (e) -- (f) node[midway,right,xshift=-2pt] {\large $1$}; \end{pgfonlayer} \end{tikzpicture} \end{center} \begin{enumerate}[label=(\alph*), itemsep=-.5em] \item Apply Kruskal's algorithm to find a minimum spanning tree of the graph. \\ \item Apply the Jarn\'{\i}k-Prim algorithm to find minimum spanning tree of the graph. \end{enumerate} For each case, include a series of pictures explaining how the minimum spanning tree is created. \end{problem} \end{document}