We classify cycles of length $k$ into three subsets:
the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
Show that, for every cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.
We classify cycles of length $k$ into three subsets:
\begin{itemize}
\item the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
\item the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
\item the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
\end{itemize}
Show that, for every cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.