% LaTeX solution template for Math 530
% -----------------------------------------------------------------

% Like most advanced LaTeX files, this one begins with a lot of
% boilerplate. You don't need to understand (or even read) most of it.
% All you need to do is fill in your name, email address,
% and the number of the pset. (Search for "METADATA" to find the place
% for this.) Then, you can go straight to the "EXERCISE 1"
% section and start writing your solutions.
% The "VARIOUS USEFUL COMMANDS" section is probably worth taking a
% look at at some point.

% If you are here to learn about LaTeX, search for
% "START READING HERE" and start reading there.

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%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
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\documentclass[paper=a4, fontsize=12pt]{scrartcl} % A4 paper and 12pt font size
\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs
\usepackage[english]{babel} % English language/hyphenation
\usepackage{amsmath,amsfonts,amsthm,amssymb} % Math packages
\usepackage{mathrsfs}    % More math packages
\usepackage{sectsty}  % Allows customizing section commands
\allsectionsfont{\centering \normalfont\scshape} % Make all section titles centered, the default font and small caps %remove this to left align section tites
\usepackage{hyperref} % Turns cross-references into hyperlinks,
                      % and defines \url and \href commands.
\usepackage{graphicx} % For embedding graphics files.
\usepackage{framed}   % For the "leftbar" environment used below.
\usepackage{ifthen}   % Used for the \powset command below.
\usepackage{lastpage} % for counting the number of pages
\usepackage[all,cmtip]{xy} % for commutative diagrams and simple graphs
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage[height=10in,a4paper,hmargin={1in,0.8in}]{geometry}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}     % This is a powerful tool to draw vector
                      % graphics inside LaTeX. In particular, you can
                      % use it to draw graphs.
\usepackage{verbatim} % For the "verbatim" environment, in which
                      % special symbols can be used freely without
                      % confusing the compiler. (And it's typeset in
                      % a constant-width font.)
                      % Useful, e.g., for quoting code (or ASCII art).

%\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)

\setlength\parindent{20pt} % Makes indentation for paragraphs longer.
                           % This makes paragraphs stand out more.

%----------------------------------------------------------------------------------------
%	VARIOUS USEFUL COMMANDS
%----------------------------------------------------------------------------------------
% The commands below might be convenient. For example, you probably
% prefer to write $\powset[2]{V}$ for the set of $2$-element subsets
% of $V$, rather than writing $\mathcal{P}_2(V)$.
% Notice that you can easily define your own commands like this.
% Caveat: Some of these commands need to be properly "guarded" when
% they occur in subscripts or superscripts. So you should not write
% $K_\CC$, but rather $K_{\CC}$.
\newcommand{\CC}{\mathbb{C}} % complex numbers
\newcommand{\RR}{\mathbb{R}} % real numbers
\newcommand{\QQ}{\mathbb{Q}} % rational numbers
\newcommand{\NN}{\mathbb{N}} % nonnegative integers
\newcommand{\FF}{\mathbb{F}} % some field
\newcommand{\Z}[1]{\mathbb{Z}/#1\mathbb{Z}} % integers modulo k
                                            % (syntax: "\Z{k}")
\newcommand{\ZZ}{\mathbb{Z}} % integers
\newcommand{\id}{\operatorname{id}} % identity map
\newcommand{\lcm}{\operatorname{lcm}}
% Lowest common multiple. For historical reasons, LaTeX has a \gcd
% command built in, but not an \lcm command. The preceding line
% rectifies that.
\newcommand{\rev}{\operatorname{rev}} % reversal of a walk
\newcommand{\powset}[2][]{\ifthenelse{\equal{#2}{}}{\mathcal{P}\left(#1\right)}{\mathcal{P}_{#1}\left(#2\right)}}
% $\powset[k]{S}$ stands for the set of all $k$-element subsets of
% $S$. The argument $k$ is optional, and if not provided, the result
% is the whole powerset of $S$.
\newcommand{\set}[1]{\left\{ #1 \right\}}
% $\set{...}$ compiles to {...} (set-brackets).
\newcommand{\abs}[1]{\left| #1 \right|}
% $\abs{...}$ compiles to |...| (absolute value, or size of a set).
\newcommand{\tup}[1]{\left( #1 \right)}
% $\tup{...}$ compiles to (...) (parentheses, or tuple-brackets).
\newcommand{\ive}[1]{\left[ #1 \right]}
% $\ive{...}$ compiles to [...] (Iverson bracket, aka truth value).
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
% $\floor{...}$ compiles to |_..._| (floor function).
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
% $\ceil{...}$ compiles to |°...°| (ceiling function).
\newcommand{\verts}[1]{\operatorname{V}\left( #1 \right)}
% $\verts{...}$ compiles to V(...) (vertex set of a graph/digraph).
\newcommand{\edges}[1]{\operatorname{E}\left( #1 \right)}
% $\edges{...}$ compiles to E(...) (edge set of a graph).
\newcommand{\arcs}[1]{\operatorname{A}\left( #1 \right)}
% $\arcs{...}$ compiles to A(...) (arc set of a digraph).
\newcommand{\lf}[2]{#1^{\underline{#2}}}
% $\lf{...1}{...2}$ compiles to $...1^{\underline{...2}}$.
% This is a notation for the falling factorial.
\newcommand{\are}{\ar@{-}}
% In an xymatrix environment, $\are$ gives an arrow without
% arrowhead. I use this to represent edges in graphs.
\newcommand{\underbrack}[2]{\underbrace{#1}_{\substack{#2}}}
% $\underbrack{...1}{...2}$ yields
% $\underbrace{...1}_{\substack{...2}}$. This is useful for doing
% local rewriting transformations on mathematical expressions with
% justifications. For example, try this out:
% $ \underbrack{(a+b)^2}{= a^2 + 2ab + b^2 \\ \text{(by the binomial formula)}} $
\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height
\newcommand{\nnn}{\nonumber\\} % Don't number this line in an "align" environment, and move on to the next line.
\newcommand{\explain}[1]{\tup{\hspace{-0.5pc}\text{\begin{tabular}{c}#1\end{tabular}\hspace{-0.5pc}}}}
% $\explain{...}$ interprets the "..." as text (you can
% nest some math inside by putting it in $...$'s as you
% would in any other text block) and puts it in parentheses.

%----------------------------------------------------------------------------------------
%	MAKING SUMMATION SIGNS ALWAYS PUT THEIR BOUNDS ABOVE AND BELOW
%	THE SIGN
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% The following are hacks to ensure that sums (such as
% $\sum_{k=1}^n k$) always put their bounds (i.e., the $k=1$ and the
% $n$) underneath and above the sign, as opposed to on its right.
% Same for products (\prod), set unions (\bigcup) and set
% intersections (\bigcap). Remove the 8 lines below if you do not want
% this behavior.
\let\sumnonlimits\sum
\let\prodnonlimits\prod
\let\cupnonlimits\bigcup
\let\capnonlimits\bigcap
\renewcommand{\sum}{\sumnonlimits\limits}
\renewcommand{\prod}{\prodnonlimits\limits}
\renewcommand{\bigcup}{\cupnonlimits\limits}
\renewcommand{\bigcap}{\capnonlimits\limits}

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%	ENVIRONMENTS
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% The incantations below define how theorem environments
% (\begin{theorem} ... \end{theorem}) and their likes will look like.
\newtheoremstyle{plainsl}% <name>
  {8pt plus 2pt minus 4pt}% <Space above>
  {8pt plus 2pt minus 4pt}% <Space below>
  {\slshape}% <Body font>
  {0pt}% <Indent amount>
  {\bfseries}% <Theorem head font>
  {.}% <Punctuation after theorem head>
  {5pt plus 1pt minus 1pt}% <Space after theorem headi>
  {}% <Theorem head spec (can be left empty, meaning `normal')>

% Environments which make the text inside them slanted:
\theoremstyle{plainsl}
  \newtheorem{theorem}{Theorem}[section]
  \newtheorem{proposition}[theorem]{Proposition}
  \newtheorem{lemma}[theorem]{Lemma}
  \newtheorem{corollary}[theorem]{Corollary}
  \newtheorem{conjecture}[theorem]{Conjecture}
% Environments that don't:
\theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{example}[theorem]{Example}
  \newtheorem{exercise}[theorem]{Exercise}
  \newtheorem{examples}[theorem]{Examples}
  \newtheorem{algorithm}[theorem]{Algorithm}
  \newtheorem{question}[theorem]{Question}
 \theoremstyle{remark}
  \newtheorem{remark}[theorem]{Remark}
\newenvironment{statement}{\begin{quote}}{\end{quote}}
\newenvironment{fineprint}{\begin{small}}{\end{small}}

%----------------------------------------------------------------------------------------
%	METADATA
%----------------------------------------------------------------------------------------
\newcommand{\myname}{Darij Grinberg} % ENTER YOUR NAME HERE
\newcommand{\mymail}{darij.grinberg@drexel.edu} % ENTER YOUR EMAIL HERE
\newcommand{\psetnumber}{3} % ENTER THE NUMBER OF THIS PSET HERE

%----------------------------------------------------------------------------------------
%	HEADER AND FOOTER
%----------------------------------------------------------------------------------------
\ihead{Math 530 Homework set \#\psetnumber} % Page header left
\ohead{page \thepage\ of \pageref{LastPage}} % Page header right
\ifoot{\myname} % left footer
\ofoot{\mymail} % right footer

%----------------------------------------------------------------------------------------
%	TITLE SECTION
%----------------------------------------------------------------------------------------
\title{	
\normalfont \normalsize 
\textsc{Drexel University, Department of Mathematics} \\ [25pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Math 530: Graph Theory, Spring 2023: \\
Homework \psetnumber\\% The assignment title
due 2023-04-28 at 11:59 PM \\
\textbf{Please solve 3 of the 6 problems!} \\
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
\author{\myname}

\begin{document}

\maketitle % This command causes the title part to be printed.

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 1}

\subsection{Problem}

Which of the exercises 1, 2, 3, 4, 5 from
homework set \#1 remain true if ``simple graph''
is replaced by ``multigraph''?

(For each exercise that becomes false, provide
a counterexample.
For each exercise that remains true, either provide
a new solution that works for multigraphs, or argue
that the solution we have seen applies verbatim to
multigraphs, or derive the multigraph case from the
simple graph case.)

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 2}

\subsection{Problem}

Let $G$ be a multigraph. Let $d > 2$ be an integer.
Assume that $\deg v > 2$ for each vertex $v$ of $G$.
Prove that $G$ has a cycle whose length is not
divisible by $d$.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 3}

\subsection{Problem}

Let $G = \tup{V, E, \varphi}$ be a multigraph that has no loops.

If $e \in E$ is an edge that contains a vertex $v \in V$, then
we let $e/v$ denote the endpoint of $e$ distinct from $v$.
(If $e$ is a loop, then this is understood to mean $v$ itself.)

For each $v \in V$, we define a rational number $q_v$ by
\[
q_v =
\sum_{\substack{e \in E; \\ v \in \varphi \tup{e}}}
\dfrac{\deg\tup{e/v}}{\deg v}.
\]
(Note that the denominator $\deg v$ on the right hand side is
nonzero whenever the sum is nonempty!)

(Thus, $q_v$ is the average degree of the neighbors of $v$,
weighted with the number of edges that join $v$ to the
respective neighbors.
If $v$ has no neighbors, then $q_v = 0$.)

Prove that
\[
\sum_{v \in V} q_v \geq \sum_{v \in V} \deg v.
\]

(In other words, in a social network, your average
friend has, on average, more friends than you do!)

\subsection{Hint}

Any two positive reals $x$ and $y$ satisfy
$\dfrac{x}{y} + \dfrac{y}{x} \geq 2$. Why, and how
does this help?

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 4}

\subsection{Problem}

Let $n$ and $k$ be two integers such that $n > k > 0$.
Define the simple graph $Q_{n, k}$ as follows:
Its vertices are the bitstrings
$\tup{a_1, a_2, \ldots, a_n} \in \set{0, 1}^n$;
two such bitstrings are adjacent if and only if they
differ in exactly $k$ bits\footnote{In other words:
Two vertices $\tup{a_1, a_2, \ldots, a_n}$ and
$\tup{b_1, b_2, \ldots, b_n}$ are adjacent if and only
if the number of $i \in \set{1, 2, \ldots, n}$
satisfying $a_i \neq b_i$ equals $k$.}.
(Thus, $Q_{n, 1}$ is the $n$-hypercube graph $Q_n$.)

\begin{enumerate}

\item[\textbf{(a)}] Does $Q_{n, k}$ have a
hamc\footnote{Recall that ``hamc'' is short for
``Hamiltonian cycle''.} when $k$ is even?

\item[\textbf{(b)}] Does $Q_{n, k}$ have a
hamc when $k$ is odd?

\end{enumerate}

\subsection{Remark}

One way to approach part \textbf{(b)} is by
identifying the set $\set{0, 1}$ with the field
$\FF_2$ with two elements. The bitstrings
$\tup{a_1, a_2, \ldots, a_n} \in \set{0, 1}^n$
thus become the size-$n$ row vectors in the
$\FF_2$-vector space $\FF_2^n$.
Let $e_1, e_2, \ldots, e_n$ be the standard
basis vectors of $\FF_2^n$ (so that $e_i$ has
a $1$ in its $i$-th position and zeroes everywhere
else). Then, two vectors are adjacent in the
$n$-hypercube graph $Q_n$ (resp. in the graph
$Q_{n,k}$) if and only if their difference
is one of the standard basis vectors
(resp., a sum of $k$ distinct standard basis
vectors).
Try to use this to find a graph isomorphism
from $Q_n$ to a subgraph of $Q_{n, k}$.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 5}

\subsection{Problem}

Let $k \in \NN$.
Let $S$ be a finite set.
The \emph{Kneser graph} $K_{S,k}$ is the simple graph
whose vertices are the $k$-element subsets of $S$,
and whose edges are the unordered pairs $\set{A, B}$
consisting of two such subsets $A$ and $B$ that
satisfy $A \cap B = \varnothing$.

Prove that this Kneser graph $K_{S,k}$ is connected
if $\abs{S} \geq 2k+1$.

\subsection{Remark}

Can the ``if'' here be replaced by an ``if and only
if''?
Not quite, because the graph $K_{S,k}$ is also
connected if $\abs{S} = 2$ and $k = 1$ (in which
case it has two vertices and one edge), or if
$\abs{S} = k$ (in which case it has only one vertex).
There might be more such ``exceptions''.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 6
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 6}

\subsection{Problem}

Let $n \geq 1$.
Let $Q_n$ be the $n$-hypercube graph, as in Spring 2022
Lecture 6 (or Spring 2023 Lecture 4).

At what vertices can a hamp\footnote{Recall that
``hamp'' is short for
``Hamiltonian path''.} of $Q_n$ end if it
starts at the vertex $00\cdots 0$ ?
(Find all possibilities, and prove that they are
possible and all other vertices are impossible.)

\subsection{Solution}

...



\horrule{0.3pt} \\[0.4cm]

% \begin{thebibliography}{99999999}                                                                                         

% \bibitem[Grinbe17]{nogra}Darij Grinberg, \textit{Notes on graph theory},
% draft of two chapters,
% 4th April 2022.\newline
% \url{https://www.cip.ifi.lmu.de/~grinberg/t/17s/nogra.pdf}

% \end{thebibliography}

\end{document}

