Hi!

I have two problems I should solve/prove in the subject of Hemiltonian cycles/paths.

1) Statement: We can create a cycle by using one pack of dominos. (domino rules: we have one brick, with two numbers (from 1 to 6) on each half of one side. we have to place the bricks next to each other, the numbers have to be the same where they "touch". E.g. we have two bricks: a 1-2 brick and a 3-2 brick. We can line them like: 1-2 and 2-3, hope you get the idea). How can I prove this statement?

2) Statement: We have a graph with more than 5 vertices. If this graph doesn't contain Hemiltonian cycle then nor does the complement of this graph. How can I prove this one?

Thanks a lot in advance!

P.s.:

Hopefully I got every mathematical expression right. English is not my native language.