another graph theory question

Thanks to everyone for their help. I seem to get the concepts of graph theory but the proofs in this class are confusing to me. There are 2 questions that i am having trouble with now....

1) If a graph G has an Euler Cycle, show that L(G) the line graph of G has a Hamilton Circuit.

I know that a graph containing an euler cycle must be connected and have each vertex with an even degree. The line graph has every edge of graph G as a vertex and is connected if they have the same end vertex in G.

I don't know why this make a Hamilton Circuit though.

2) This one says suppose a set I of k vertices in a graph, G, is chosen so that no pair of vertices in I are adjacent. Then for each x in I, deg(x) -2 of the edges incident to x will not be used in a Hamilton Circuit.

Then, summing over all vertices in I, we have e'= Sum [x element of I] (deg(x) - 2) which is equivalent to {Sum with x element of I of (deg(x))} -2k.

The problem states:

Let v and e be the numbers of vertices and edges in G respectively. Show that if e - e' < v, then G can have no Hamilton Circuit.

I also have to show why it is true only when I is a set of non-adjacent vertices.

Any help is appreciated.

Thanks

Ernie