A neighborly triangulation is a triangulation by a simplicial complex in which every pair of
distinct vertices is an edge. Thus its edge graph
is a complete graph. More generally, a complex
is 
-neighborly
when every set of at most 
vertices is a simplex.
Unqualified neighborliness in this setting means 2-neighborliness.
The boundary complex of the Császár polyhedron is a neighborly triangulation of the torus.
Its seven vertices and 21 edges give the complete edge
graph 
. Kühnel's nine-vertex triangulation
of the complex projective plane is 3-neighborly,
so every triple of vertices is a triangular face (Kühnel
and Lutz 2000).
For a closed orientable surface, neighborliness is equivalent to tightness over any field. In higher dimensions, neighborliness is necessary for a tight triangulation but is not sufficient.
