Question

Consider a graph where V(G)={1, 2, 3, 4} and E(G)=[{1,2}, (1,2), {1,4}, {2,3}, {3,4}, {3,4}]. How many Hamilton cycles does it have?

Answer 1

There are 2 ways to choose the direction (clockwise or counter-clockwise). There are two ways to choose an edge between 1,2 and 2 ways to choose an edge between 3,4. I.e. there are $2\cdot 2\cdot 2=8$ cycles.