Exercise 4, § 4

Let E be a finite set with $n \geqslant 2$ elements, and let C be a subset of $\mathrm{E} \times \mathrm{E}$ such that, for each pair $x, y$ of distinct elements of E, exactly one of the two elements $(x, y)$, $(y, x)$ of $\mathrm{E} \times \mathrm{E}$ belongs to C. Show that there exists a mapping $f$ of the interval $[1, n]$ onto E such that $(f(i), f(i + 1)) \in \mathrm{C}$ for $1 \leqslant i \leqslant n - 1$ (use induction on $n$).