I was asked to draw a Hamilton graph having a Hamilton cycle that traverse an edge more than once.
My first impression of this question was: what? I mean if we are not allowed to visit a vertex more than once doesn't it mean we are not allowed to traverse an edge more than once. But then I thought of the following solution. Consider the graph
Then this is a Hamilton graph with a Hamilton cycle $$ v_1,e_1,v_2,e_1,v_1 $$
My question: First, am I right? Can you draw a different Hamilton graph having a Hamilton cycle that traverse an edge more than once? By a different I mean a graph that does not contain the given graph above as its subgraph?