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\begin{remark*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $\chrompoly_G(t)$ does carry other
      information about $G$. For example, it turns
      out that
      \[ \chrompoly_G(t) = t^n - mt^{n - 1} +
      \left( {n \choose 2} - \text{\# triangles in
      $G$} \right) t^{n - 2} - \cdots \]
    \item As $\chrompoly_G$ is a polynomial, we
      can talk about $\chrompoly_G(t)$ for any
      real (or complex) $t$.
    \item The \nameref{4_col_thm} says: $G$
      \gls{planar} $\implies$ $\chrompoly_G$ does
      not have a root at $4$. No such `polynomial'
      proof of the \nameref{4_col_thm} is known.
      It \emph{is} known that $\chrompoly_G(2 +
      \phi) \neq 0$, where $\phi = \frac{1 +
      \sqrt{2}}{2}$ ($2 + \phi$ is approximately
      $3.6$).
  \end{enumerate}
\end{remark*}

\subsection{Edge Colourings}

\begin{flashcard}[edge-colouring-defn]
\begin{definition*}[Edge colouring]
  \glsnoundefn{k_edge_col}{$k$-edge-colouring}{$k$-edge-colourings}
  \cloze{
  A \emph{$k$-edge-colouring} of a \gls{graph} $G$
  is a function $c : \eset(G) \to [k]$ such that
  $c(e) \neq c(e')$ whenever $e \neq e'$ share a
  vertex.
  }
\end{definition*}
\end{flashcard}

\begin{flashcard}[edge-chrom-num-defn]
\begin{definition*}[Edge-chromatic number]
  \glssymboldefn{edge_chrom_num}{$\chi'$}{$\chi'$}
  \glspropdefn{edge_chrom_num}{edge-chromatic
  number}{\gls{graph}}
  \cloze{
  The \emph{edge-chromatic number} of
  \emph{chromatic index} of $G$, written
  $\chi'(G)$, is the least $k$ such that $G$ has a
  \gls{k_edge_col}.
  }
\end{definition*}
\end{flashcard}

\begin{example*}
  \[ \edgechromnum(\cycle_n) = \begin{cases}
    2 & \text{if $n$ even} \\
    3 & \text{if $n$ odd}
  \end{cases} \]
\end{example*}
\vspace{-1em}

Note that $\edgechromnum(G) =
\chromnum(\linegraph(G))$ (but sadly this is not
much help).

Can have $\edgechromnum, \chromnum$ far apart, for
example $\bipartite_{1, n}$ has
$\chromnum(\bipartite_{1, n}) = 2$,
$\edgechromnum(\bipartite_{1, n}) = n$.

Always have $\edgechromnum(G) \ge \maxdeg(G)$ (and
not always equal, for example $\cycle_{\text{odd}}$).
Also, $\edgechromnum(G) \le 2\maxdeg(G) - 1$ (run
greedy on any ordering).

Calculating $\edgechromnum(\complete_n)$ is an
exercise on \es{3}. This \gls{graph} is
interesting because it is not entirely obvious how to calculate
the \gls{edge_chrom_num}, despite the fact that
the \gls{graph} has such a nice structure.

Remarkably:

\begin{flashcard}[vizings-thm]
\begin{theorem}[Vizing's Theorem]
  % Theorem 7
  \label{vizings_thm}
  \cloze{
  For any \gls{graph} $G$, $\edgechromnum(G) =
  \maxdeg(G)$ or $\edgechromnum(G) = \maxdeg(G)$.
  }
\end{theorem}

\begin{proof}
  \cloze{
  We'll show that every graph has a
  \edgecol{(\maxdeg + 1)}.

  Induction on $\edges(G)$, $\edges(G) = 0$
  trivial.

  Given $G$, $\edges(G) > 0$: choose any $e \in
  E$, and have \edgecol{(\maxdeg + 1)} of $G - e$
  (induction), say $e = xy$. Note that every
  \gls{vertex} has an unused colour (as number of
  colours is greater than max degree): say colour
  $c$ at $x$ and $c_1$ at $y$. Without loss of
  generality $c \neq c$, else done (use colour $c$
  for $x$).
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/a0ef2230fe6341ee.png}
  \end{center}
  Choose a maximal sequence $y_1, \ldots, y_k$ of
  distinct \glspl{neighbour} of $x$ such that some
  colour $c_i$ is missing at $y_i$ and $xy_i$ has
  colour $c_{i - 1}$ (each $2 \le i \le k$).
  (Exists since the \gls{graph} is finite). Must
  have stopped at $y_k$ because either $c_k$ does
  not occur at $x$ or $c_k = c_j$ for some $j <
  k$.

  If $c_k$ does not occur at $x$, can give $xy_i$
  colour $c_i$ (all $1 \le i \le k$), giving a
  legal colouring of $G$.

  If $c_k = c_j$ for some $j < k$ then without
  loss of generality $j = 1$, by giving edge
  $xy_i$ (each $i < j$) colour $c_i$ -- leaving
  $xy_j$ as the uncoloured edge.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/865c531798c94abd.png}
  \end{center}
  If no $c-c_1$ path from $y_1$ to $x$: swap
  colours $c$ and $c_1$ on the $c-c_1$ component
  of $y_1$. Now $c$ is missing at $y_1$ (by our
  swap) and $c$ is still missing at $x_1$, so can
  colour $xy_1$ with colour $c$. So may assume
  there exists $c-c_1$ path from $y_1$ to $x$.

  If no $c-c_1$ path from $y_k$ to $x$: swap
  colours $c$ and $c_1$ on the $c-c_1$ component
  of $y_k$. Now $c$ is missing at $y_k$ and at
  $x$. So done by giving $xy_i$ colour $c_i$ ($1
  \le i \le k - 1$) and give $xy_k$ colour $c$. So
  may assume that there also exists $c-c_1$ path
  from $y_k$ to $x$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/790bd8fb785d4e9e.png}
  \end{center}
  Let $H$ be the $c-c_1$ component of $x$. Then
  $\maxdeg(H) \le 2$, and $H$ \gls{connected}, so
  $H$ is a \gls{path} or a \gls{cycle}. But
  $\vdeg_H(x) = \vdeg_H(y_1) = \vdeg_H(y_k) = 1$,
  contradiction.
}
\end{proof}
\end{flashcard}

\subsection{Graphs on Surfaces}

We know that $\chromnum(G) \le 5$ (actually $\le
4$) for any \gls{graph} drawn in the plane, or
equivalently on a sphere. What happens on other
surfaces?

\begin{flashcard}[surface-of-genus-g-defn]
\begin{definition*}[Surface of genus $g$]
  \glsnoundefn{surf}{surface}{surfaces}
  \cloze{
  For $g = 0, 1, 2, \ldots$ the \emph{surface of
  genus $g$} (or the \emph{compact orientable
  surface of genus $g$}) consists of the sphere,
  with $g$ handles attached:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/10998a9db2d5491c.png}
  \end{center}
  }
\end{definition*}
\end{flashcard}
\vspace{-1em}

Known that for planar $G$:
\begin{align*}
  n - m + f
  &= 2
  &&\text{($G$ \gls{connected})} \\
  n - m + f
  &\ge 2
  &&\text{(general $G$ -- add \glspl{edge} to make
  $G$ \gls{connected})}
\end{align*}
What about on a torus?
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/6d36b1000b134db1.png}
\end{center}
\glssymboldefn{euler_char}{$E$}{$E$}
\glspropdefn{euler_char}{Euler
characteristic}{\gls{graph}}
\textbf{Fact:} for any \gls{graph} on surface of
genus $g$, $n - m + f \ge 2 - 2g$. We call $2 -
2g$ the \emph{euler characteristic} of the
surface, written $\eulerchar$.