Key point 1.16 A Hamiltonian graph contains a cycle that passes through every vertex exactly once, apart from starting and finishing at the same vertex. The cycle is called a Hamiltonian cycle. There will usually be edges in the graph that are not used in the cycle.
Common error
WORKED EXAMPLE 1.17
PL E
Be careful not to confuse Hamiltonian graphs (which contain a closed trail that uses every vertex once) and Eulerian graphs (which contain a closed trail that uses every edge once).
In chess, a knight moves two squares horizontally followed by one square vertically or two squares vertically followed by one square horizontally.
M
A classic chessboard puzzle, known as the knight’s tour, asks whether a knight can visit every square of an 8 × 8 chessboard and whether this can be done with the knight finishing back where it started.
SA
The centre of each square on the chessboard is a vertex. The solution requires a Hamiltonian cycle made up of edges that are knight’s moves. There are several possible solutions.
WORKED EXAMPLE 1.18
Draw a Hamiltonian cycle on the edges of a cube. Each corner of the cube is a vertex. The solution requires a Hamiltonian cycle made up of edges of the cube. A possible solution is shown.
Original material © Cambridge University Press 2018