The Hauptvermutung is the statement that any two PL structures on a topological space have a common refinement. It is false in general, but (I think) true for some low dimensional manifolds.

The special case I'll consider is when the topological space is a smooth 2-dimensional manifold. In this case, the Hauptvermutung (roughly) states that for any two embeddings of two graphs into the manifold (such that the resulting faces are homeomorphic to disks), there is a graph (with an embedding) that contains both graphs as a subgraph that preserves that PL structure (cue the hand-waving).

It's "obvious" that if you take the union of the two graphs (again, embedded into the manifold) and add all the necessary points (such as the points at intersections of edges of the graphs), that you'll get a refinement common to both graphs (bar some subtle cases, like when the two graphs don't intersect)

In this answer, it's stated that this proof is wrong for "fractal" reasons. Does this mean that the proof still goes through if we restrict to finite graphs? If not, why?

This question is motivated by a problem posed a year or two back in my Differential Geometry class. The problem required that I prove that the Euler Characteristic (on a smooth, compact surface), defined by the quantity $V-E+F$ for any embedding of a graph, is well defined. I used essentially the proof above to construct a common refinement of any two graphs that preserved the quantity $V-E+F$. Since that particular homework was never returned, I never found out if my proof was correct, so now I'm wondering if I missed something.

**EDIT:** Since this approach clearly doesn't work (thanks to Alex Degtyarev in the comments), I might ask if there's any other way of proving that the Euler characteristic for compact surfaces defined in this way is well-defined. I know you can just use the homology groups (which is the preferred approach), but since we didn't mention homology in my class, I suspect there's a more elementary method of proof.