On graph imbedding genus clarification 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.


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*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?). 
 A: (Not an answer, yet too long for the comment box, and thought by me to be helpful for the OP)
I do not understand even your question 1.
I had too hard a time parsing your question 1., which has at least grammatical issues, and now I would like to at least understand what you are asking. (If others think this question is clear, please tell me.)
Supposedly you mean something along the lines of:

Does there exist, for every genus-$g$ geometric finite graph $G=(V,E)$ an imbedding $\eta\colon G\rightarrow \mathbb{T}_0\#\mathbb{T}_1\#\dotsm\#\mathbb{T}_{g-1} =:S$ (connected sum of tori, themselves embedded in the usual way into $\mathbb{R}^3$) and a function $h\colon E\rightarrow g$ (taking $g$ to be a finite ordinal)  such that for each $e\in E$,

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*the topological arc $\eta(e)$ has WHATRELATION to the summand $\mathbb{T}_{h(e)}$, and


*the closed curve $C$ obtained by connecting the ends of $\eta(e)$ by a geodesic [do you mean that?] in $S$, regardless of whether said geodesic intersects other arcs of the embedding [which it seems generically it does], is not a separating curve in $S$?

Would you please clarify what WHATRELATION should be here. It is your question.
Also, would you please clarify

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*whether when saying "handles" you mean that you are only interested in orientable surfaces with genus $\geq 2$?
Would you accept a counterexample to (an interpretation of) question 1. on the torus, which of course only has only one handle?

You might also (if you first work on your questions to make them conveniently understandable, and if you first,before emailing, have a look at the visualizations he has created of such things) try to ask Professor Bokowski at TU Darmstadt. I take it that he is someone actively working on problems relevant to you question.
