# 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.

1. 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)?

2. 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?).

• Can we rephrase the question as follows? "If $G\to\Sigma_g$ is a minimal embedding, then is the following the case? There exists a non-separating collection of disjoint simple closed curves $\gamma_1,\ldots,\gamma_g$ on $\Sigma_g$ so that the vertices of $G$ are disjoint from the $\gamma_i$, and each edge of $G$ intersects $\bigcup_i\gamma_i$ in at most one point." – HJRW Aug 7 '17 at 12:02

(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 $e\in E$,

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

2. 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.

• 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?
• updated query hope it is clear. I just want to know if there is an example of a genus $g$ graph where 'every' embedding of the graph in to a surface of genus $g$ needs at least one edge to utilize two handles to make the embedding work. – Brout Aug 7 '17 at 7:30