There is a vast literature on embeddings of graphs into surfaces. I am interested in embeddings of graphs that belong to the given homotopy class. Here is the precise formulation.
I have two finite graphs $\Gamma, \Gamma'$ and a homotopy equivalence $f: \Gamma\to \Gamma'$. I know that there exists an embedding $\iota: \Gamma'\to S$, where $S$ is a fixed closed oriented surface. I am interested in finding an embedding $j: \Gamma \to S$ so that $j$ is homotopic to $\iota\circ f$. This is, of course, impossible without further restrictions on topology of $\Gamma$, since one can take, for instance, $\Gamma=K_5$, $\Gamma'$ to be the rose with $6$ petals and $S=S^2$. Edit: Therefore, let us assume, say, that
$$ (*) \quad \chi(\Gamma) \ge \chi(S)-1 $$
and that $\iota_*: \pi_1(\Gamma')\to \pi_1(S)$ is surjective.
Question 1. Does the inequality (*) together with surjectivity assumption above imply that $\iota\circ f$ is homotopic to an embedding for every $f$ and every embedding $\iota: \Gamma'\to S$?
In fact, the inequality (*) seems to be way too generous. Assuming that $\Gamma=K_n$ and taking into account the formula for the genus of $K_n$, one arrives to:
Question 2. Suppose that, $\chi(S)<0$ and replace (*) with $$ \chi(\Gamma)> 3 \chi(S). $$ Does it follow that $\iota\circ f$ is homotopic to an embedding?