Let $S$ be a surface obtained by identifying the sides of a regular hexagon in pairs.
I want to show $\chi(S) > -1 $.
I can see how we can obtain surfaces with $\chi(S) = 0,1,2$ but I think I'm missing something for this inequality.
We start of with 6 vertices, 6 edges, 1 face. Whenever I identify a pair of edges it seems to "remove" edges and vertices but I haven't made this into an argument.