C
3
Use Euler’s formula for the planar graph K to find the number of regions.
4
Show that this graph is planar by drawing it after it has been manipulated so that no edges cross.
5
Show that these graphs are not isomorphic.
SA
M
PL E
4
6
Use Euler’s formula to find the number of regions in the planar graph described by this adjacency matrix. 0
1
0
1
0
1
⎜1 ⎜ ⎜0 ⎜ ⎜ ⎜1 ⎜ ⎜0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
1
0
0⎟ ⎟ 1⎟ ⎟ ⎟ 1⎟ ⎟ 1⎟
0
1
1
1
0
⎛
⎝
7
1
⎞
⎠
A standard dice is modelled as a planar graph with vertices labelled 1, 2, 3, 4, 5 and 6. The faces of the dice are represented by the vertices of the graph and edges connect vertices that correspond to adjacent faces, but not vertices that correspond to opposite faces. Original material © Cambridge University Press 2018