3 Fig. 3 shows a graph representing the seven bus journeys run each day between four rural towns. Each directed arc represents a single bus journey.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee39642f-f323-4614-a02a-4500199626de-4_317_515_392_772}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{figure}
- Show that if there is only one bus, which is in service at all times, then it must start at one town and end at a different town.
Give the start town and the end town.
- Show that there is only one Hamilton cycle in the graph.
Show that, if an extra journey is added from your end town to your start town, then there is still only one Hamilton cycle.
- A tourist is staying in town B. Give a route for her to visit every town by bus, visiting each town only once and returning to B .
Section B (48 marks)