There's no need to assume exactly 3 hexagons meet at each vertex: if you only assume at least 3 hexagons meet at each vertex, then 6N >=3V, and we get that the

There's no need to assume exactly 3 hexagons meet at each vertex: if you only assume at least 3 hexagons meet at each vertex, then 6N >=3V, and we get that the Euler characteristic is <=0, which is still a contradiction. Of course if you allow vertices with only two hexagons meeting at them you can make a sphere: for example, take two hexagons and glue them around the edges, then stuff the middle to get a hexagonal cushion.