Given a graph the minimum genus $g$ is the minimum number of handles needed so that there an imbedding of the graph on the surface with no edge crossings.

If the graph is of genus $g$ then is there

**always**an imbedding such that every edge traverses at most one handle (traverses like how a non-separating circle would if both ends of the edges are joined together)?If not then is there

**always**an imbedding into a surface of genus $g'$ larger than $g$ such that every edge traverses at most one handle(traverses like how a non-separating circle would if both ends of the edges are joined together)? What is the name for this genus $g'$?

Note there are two ways an edge traverses a handle. I want each edge to use only one handle at most once.

I think 1. is likely the case going with interpreting homology group ($\Bbb Z^{2g}$) as totality of all the handles used by edges at most once (two edges can use same handle but I want one handle use per edge) but I do not know of any proof that directly stresses this. And so I think $g'=g$ but I do not know with certainty (is there any reference?).