Networks/graph theory

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Without taking your pencil off the paper or going over a line or passing through one of the points twice, can you follow each of the networks?

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?