I'm reading Thurston's article "Shapes of polyhedra and triangulations of the sphere." In the introduction he claims the following:



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> "${}^{(1)}$There are procedures to refine and modify any triangulation of a surface until every vertex has either 5, 6 or 7 triangles around it, or with more effort, ${}^{(2)}$so that there are only 5 or 6 triangles if the surface has positive Euler characteristic, only 6 triangles if the surface has zero Euler characteristic, or only 6 or 7 triangles if the surface has negative Euler characteristic..."
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I divide the quote into two claims. I proved the first claim, but I'm stuck with the second one. There must be a global argument I'm not seeing. Does anyone know how to do it?



Refining a triangulation
Suppose that $S$ is a closed, connected surface with negative Euler characteristic. Suppose that $T$ is a triangulation of $S$.


Define "refine" to mean "replace each triangle by four triangles" (so that edge midpoints become vertices of valence six). This does not improve any of the vertices of "concentrated positive or negative curvature", but it does isolate them.


Suppose that there are no vertices of degree five or lower. Refine as above to isolate all vertices of degree greater than six. Suppose that the vertex $v$ has degree eight or higher. We "split" at $v$ - we choose two edges $e$ and $e'$ at $v$, cut $T$ along $e \cup e'$ and insert a pair of triangles. This increases the number of triangles by two, the number of vertices by one, and the number of edges by three. The new vertices $u$ and $u'$ have degree less than that of $v$. Also, two old vertices of degree six will now have degree seven. Repeat.




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As a bit of motivation: the original triangulation gives a metric on $S$. Refinement has the effect of scaling the metric up, which "brings the curvature down". Cutting and inserting triangles disperses concentrated negative curvature (at a vertex) into adjacent (almost) flat regions.




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In general, if there are vertices of degree less than six, we refine and then pair them with vertices of higher degree, and (carefully!) cancel curvature.


In the case of Euler characteristic zero, after pairing in this way, only vertices of degree six remain.


In the case of positive Euler characteristic, we have some vertices of degree five (either six or twelve) left at the end.


