Graph embedding into a surface For example, let's consider a $K_{5}$ (complete graph on 5 vertixes) and a torus, which is defined as $S^{1} \times S^{1}$. How to build a continous embedding $f:K_{5} \rightarrow \mathbb{T}^{2}$?
We can prove that if it's possible to draw a graph $(V, E)$ on a sphere with $g$ handles, then the following Euler's inequality holds $V-E+F \ge 2-2g$. But how to prove(maybe, relying on the Euler characteristic) the existense of such embedding for a given graph and surface? Probably, it would be less complicated to build an exact embedding, but how to do it without just drawing a torus (for example) and a graph on it (of course, which can be accepted as a proof because of lack of rigour)?
Any help would be much appreciated. 
 A: Explicitly constructing the embedding is pretty easy: take (in $[-\pi, \pi] \times [-\pi, \pi]$) the vertices and edges of the  square whose vertices are $(\pm \frac{\pi}{2}, \pm \frac{\pi}{2})$, and the lines $x = \pm y$. These intersect at an additional vertex (at $(0,0)$), and when these lines together with the original square's vertices and edges, are considered as points of the torus, they form an embedding of $K_5$. 
I give this example partly because many of my students seem afraid of explicitly constructing examples, and like to use big theorems to prove stuff when it can often be more easily understood by a clearly described example. (And such examples can indeed by rigorous). Constructing examples is a great way to test conjectures as well. I've answered many questions on math.stackexchange about $\mathbb R^n$ by constructing fairly simply counterexamples in, say, $\mathbb R^1$ or $\mathbb R^2$. 
I don't entirely understand the other part of your question --- for a cellular decomposition of a sphere with $g$ handles into $V$ vertices, $E$ edges, and $F$ disk-like faces , $V - E + F = 2 - 2g$ exactly. But it's not clear how this is related to the $V$ and $E$ for your graph. 
