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\section*{Corrigendum}

The following comment appears after the proof of Theorem 12 on
p.~14:

\begin{displayquote}
``The computer searches referred to in the proof of Theorem~12
can be extended to show that there are exactly two snarks with
oddness $4$ on $28$ vertices—those displayed in Figure~2.''
\end{displayquote}

As noted by Goedgebeur in \cite{G}, there are three snarks with
oddness $4$ on $28$ vertices. The third, missing, snark is
displayed in Figure~\ref{fig}.

\begin{figure}[h]
\begin{center}
\includegraphics{fig-1}
\end{center}
\caption{The third snark of oddness $4$ on $28$ vertices.}\label{fig}
\end{figure}

The removal of one of the $2$-edge-cuts from the snark in
Figure~\ref{fig} produces two components, $G_1$ on $18$
vertices and $G_2$ on $10$ vertices. Let $G'_1$ denote the
graph obtained from $G_1$ by adding an edge $e_1$ that connects
the two vertices of degree two. It turns out that $e_1$ is a
special edge (as defined in the proof of Theorem~12) and thus,
by the argumentation in the proof of Theorem~12, this graph may
have (and, indeed, has) oddness $4$. Due to an error in our
computer search, we did not find $G'_1$ among the graphs on
$18$ vertices containing a special edge.

Although Theorem~12, as stated, is not false and its proof is
correct, the following formulation is more precise.

\begin{maintheorem}
The smallest snark with oddness $4$ has $28$ vertices. There
are exactly three such snarks, two with cyclic connectivity $2$
and one with cyclic connectivity $3$.
\end{maintheorem}

\begin{thebibliography}{9}
\bibitem{G} J. Goedgebeur, On the smallest snarks with oddness
    $4$ and connectivity $2$, arXiv: 1710.00757.
\end{thebibliography}

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