Cycles of length $k$ are classified 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 any cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.
Cycles of length $k$ are classified 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 any cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.