3 The adjacency graph of a cube
\includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_106_108_214_735}
is shown. Vertices on the graph represent faces of the cube. Two vertices are connected by an arc if the corresponding faces of the cube share an edge.
\includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_401_464_246_1334} The second graph is the complement of the adjacency graph, i.e. the graph that consists of the same vertices together with the arcs that are not in the adjacency graph.
\includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_403_464_737_1334} Throughout this question we wish to colour solids so that two faces that share an edge have different colours. The second graph shows that the minimum number of colours required for a cube is three, one colour for the top and base, one for the front and back, and one for the left and right.

1. Draw the adjacency graph for a square-based pyramid, and draw its complement. Hence find the minimum number of colours needed to colour a square-based pyramid.
   \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_161_202_1434_1548}
2. (A) Draw the adjacency graph for an octahedron, and draw its complement.
   (B) An octahedron can be coloured using just two colours. Explain how this relates to the complement of the adjacency graph.
   \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_227_205_1731_1548}