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% 05/10/2023 9AM

% Course overview
% CH1: INTRODUCTION
% CH2: CONNECTIVITY & MATCHINGS
% CH3: EXTREMAL PROBLEMS
% CH4: COLOURINGS
% CH5: RAMSEY THEORY
% CH6: RANDOM GRAPHS
% CH7: ALGEBRAIC METHODS

\mychapter{Introduction}

\section{General Words of Advice}

This course is meant to be quite elegant.
Understanding the proofs in this course is
essential. Exam questions will be about problems
where you apply proof methods seen in lectured
proofs.

Proofs will be elegant, and will generally not
involve calculations. We will use matrices
sometimes (and in particular we may talk about
eigenvalues) but you should never have to
calculate a determinant of a matrix on any example
sheet or exam question.

\subsection{Basic Definitions}

\begin{flashcard}[graph-defn]
\begin{definition*}[Graph]
  \glsnoundefn{graph}{graph}{graphs}
  \cloze{
  A \emph{graph} is a pair $(V, E)$, where $V$ is
  a set and $E$ is a \fcemph{subset} of $V^{(2)} = \{(x, y)
  : x, y \in V, x \neq y\}$, the \fcemph{unordered
  pairs} from $V$.

  \glsnoundefn[graph]{vertex}{vertex}{vertices}
  \glsnoundefn[graph]{edge}{edge}{edges}
  $V$ is the \fcemph{vertex set} of $G$ and $E$ is
  the \fcemph{edge set} of $G$.
  }
\end{definition*}
\end{flashcard}

\begin{example*}
  The graph below has $V = \{x_1, x_2, x_3, x_4,
  x_5\}$, $E = \{\{x_1, x_2\}, \{x_1, x_4\},
  \{x_2, x_3\}, \{x_3, x_4\}, \{x_4, x_5\}\}$.
  \begin{center}
      \includegraphics[width=0.6\linewidth]
      {images/494d5dfa635911ee.png}
  \end{center}
\end{example*}

\begin{flashcard}[graph-order-defn]
\begin{definition*}[Order of a graph]
  \cloze{The \emph{order} of $G$ is $|G| = |V|$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[graph-size-defn]
\begin{definition*}[Size of a graph]
  \cloze{The size of $G$ is $e(G) = |E|$.}
\end{definition*}
\end{flashcard}

\subsubsection*{Notes}

\begin{enumerate}[(1)]
  \item No self loops (edges starting and ending
    at the same vertex).
  \item No multiple edges (between every pair of
    vertices, there are exactly 0 or 1 edges).
  \item No directed edges.
\end{enumerate}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
  \item \glsnoundefn{empty_graph}{empty
    graph}{empty graphs}
    \glssymboldefn{empty_graph}{$E$}{$E$}
    The \emph{empty graph} $E_n$: $V = \{x_1,
    \ldots, x_n\}$, $E = \emptyset$. So $|E_n| =
    n$, $e(E_n) = 0$.
    \begin{center}
        \includegraphics[width=0.3\linewidth]
        {images/848892b2635a11ee.png}
    \end{center}
    \glsnoundefn{complete_graph}{complete
    graph}{complete graphs}
    \glssymboldefn{complete_graph}{complete
    graph}{complete graph}
  \item The \emph{complete graph} $K_n$: $V =
    \{x_1, \ldots, x_n\}$, $E = V^{(2)}$, so
    $|K_n| = n$, $e(K_n) = {n \choose 2}$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/91055db8635a11ee.png}
    \end{center}
  \item The \emph{path of length $n$}, $P_n$: $V =
    \{x_1, \ldots, x_{n + 1}\}$, $E = \{x_1 x_2, x_2
    x_3, \ldots, x_n x_{n + 1}\}$, so $|P_n| = n +
    1$, $e(P_n) = n$.
    \begin{center}
        \includegraphics[width=0.3\linewidth]
        {images/9bcd61f0635a11ee.png}
    \end{center}
    \glsnoundefn{cycle}{cycle}{cycles}
    \glssymboldefn{cycle}{$C$}{$C$}
  \item The \emph{$n$-cycle} $C_n$: $V = \{x_1,
    \ldots, x_n\}$, $E = \{x_1 x_2, x_2 x_3,
    \ldots, x_{n - 1} x_n, x_n x_1\}$, so $|C_n| =
    n$, $e(C_n) = n$.
    \begin{center}
        \includegraphics[width=0.3\linewidth]
        {images/d699ed9e635a11ee.png}
    \end{center}
\end{enumerate}

\begin{flashcard}[graph-isomorphism]
\begin{definition*}[Graph Isomorphism]
  \cloze{
  Say graphs $G = (V, E)$ and $H = (V', E')$ are
  \emph{isomorphic} if there exists bijection $f :
  V \to V'$ such that $xy \in E \iff f(x)f(y) \in
  E'$.
  }
\end{definition*}
\end{flashcard}

\begin{example*}
  \phantom{}
  \begin{center}
      \includegraphics[width=0.6\linewidth]
      {images/6c9c9a76635b11ee.png}
  \end{center}
\end{example*}

\begin{flashcard}[subgraph-defn]
\begin{definition*}[Subgraph]
  \glsnoundefn{subgraph}{subgraph}{subgraphs}
  \cloze{
  Say $H$ is a \emph{subgraph} of $G$ if $V'
  \subset V$, $E' \subset E$.
  }
\end{definition*}
\end{flashcard}

\begin{notation*}
  For $G = (V, E)$ and $xy \in E$, write $G - xy =
  (V, E \setminus \{x, y\})$.
\end{notation*}

\begin{notation*}
  For $G = (V, E)$ and $xy \in V^{(2)} \setminus
  E$, write $G + xy = (V, E \cup \{x, y\})$.
\end{notation*}

\begin{flashcard}[neighbous-defn]
\begin{definition*}[Neighbours]
  \glsadjdefn{adj_v}{adjacent}{vertices}
  \glsnoundefn{neighbour}{neighbour}{neighbours}
  \cloze{If $xy \in E$, say $x$, $y$ are
  \emph{adjacent} or \emph{neighbours}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[neighbourhood-defn]
\begin{definition*}[Neighbourhood]
  \glssymboldefn{nbd}{$\Gamma$}{$\Gamma$}
  \cloze{For $x \in G$, the \emph{neighbourhood} of $x$
  is $\Gamma(x) = \{y \in G : xy \in E\}$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[vertex-degree-defn]
\begin{definition*}[Vertex degree]
  \glspropdefn{v_deg}{degree}{vertex}
  \glssymboldefn{v_deg}{degree}{vertex}
  \cloze{For $x \in G$, the \emph{degree} of $x$ is $d(x)
  = |\Gamma(x)|$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[degree-sequence-defn]
\begin{definition*}[Degree Sequence]
  \cloze{
  If $V = \{x_1, \ldots, x_n\}$ then the
  \emph{degree sequence} of $G$ is $d(x_1), \ldots,
  d(x_n)$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[min-max-degree]
\begin{definition*}[Minimum and Maximum Degree]
  \glssymboldefn{min_deg}{$\delta$}{$\delta$}
  \glssymboldefn{max_deg}{$\Delta$}{$\Delta$}
  \cloze{
  $G$ has \emph{maximum degree}
  $\Delta(G) = \max\{d(x_1), \ldots, d(x_n)\}$ and
  \emph{minimum degree} $\delta(G) = \min\{d(x_1),
  \ldots, d(x_n)\}$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[k-regular-defn]
\begin{definition*}[Regular Graph]
  \glsadjdefn{regular}{regular}{\gls{graph}}
  \cloze{
  If the degree of each vertex in $G$ is $k$, then
  we say that $G$ is \emph{$k$-regular} (or just
  \emph{regular}).
  }
\end{definition*}
\end{flashcard}
\vspace{-1em}

\noindent
For example, $C_n$ is $2$-regular, and $K_n$ is
$(n - 1)$-regular. In this course, unless
otherwise stated, $V$ is finite (we will study
infinite graphs briefly).

\subsection{Connectedness}

\begin{flashcard}[path-defn]
\begin{definition*}[Path]
  \glsnoundefn{path}{path}{paths}
  \cloze{
  For $x, y \in G$, an $x-y$ path is a sequence
  $x_1, \ldots, x_k$ ($k \ge 1$) of distinct
  vertices of $G$ such that $x_1 = x$, $x_k = y$
  and $x_i x_{i + 1} \in E \forall i$. Its
  \emph{length} is $k - 1$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[connected-squiggle]
\begin{notation*}
  Write $x \sim y$ if \cloze{there exists a path
  from $x$ to $y$ (an $x-y$ path).}
\end{notation*}
\end{flashcard}
\vspace{-1em}

\noindent
Note that this is an equivalence relation.

\begin{remark*}
  Given an $x-y$ path and a $y-z$ path,
  concatenating them does \emph{not} necessarily
  give an $x-z$ path.
  \begin{center}
      \includegraphics[width=0.6\linewidth]
      {images/a71df82c635e11ee.png}
  \end{center}
  However, it always \emph{contains} an $x-z$ path
  -- indeed, if we set $i$ to be the least $1 \le
  i \le k$ such that there exists $k \le j \le l$
  with $x_j = x_k$, then $x_1, \ldots, x_i, x_{j +
  1}, \ldots, x_l$ is an $x-z$ path.
\end{remark*}

\begin{flashcard}[connected-components-defn]
\begin{definition*}[Component]
  \glsnoundefn{conn_comp}{component}{components}
  \cloze{
  The equivalence classes of $\sim$ are the
  \emph{components} of $G$. The component $[x]$ of
  a vertex is $[x] = \{y : \exists \text{$x-y$
  path in $G$}\}$.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[connected-graph-defn]
\begin{definition*}[Connected Graph]
  \glsadjdefn{connected}{connected}{graph}
  \cloze{Say $G$ is connected if $\forall x, y \in G$,
  there exists an $x-y$ path in $G$.

  In other words, has one component \fcemph{or has
  no vertices}.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[walk-defn]
\begin{definition*}[Walk]
  \glsnoundefn{walk}{walk}{walks}
  \cloze{An $x-y$ walk is a sequence $x_1, \ldots, x_k$
  ($k \ge 1$) such that $x_1 = x$, $x_k = y$ and
  $x_i x_{i + 1} \in E \forall i$.} 
\end{definition*}
\end{flashcard}
\vspace{-1em}

\noindent
So a walk is a path where we allow repeats.

\begin{remark*}
  There exists a walk from $x$ to $y$ if and only if
  there exists a path from $x$ to $y$.
\end{remark*}