Exponential Families of Non-Isomorphic Triangulations of Complete Graphs

Recursive constructions for triangulations

“…We may use the construction with m = 3 and k = 2 by making use of the two differently labelled K 3,3,3 embeddings given in [1]. This gives 2 n 2 differently labelled 2to-embeddings of K 3n,3n,3n .…”

“…Writing w for 3n, we may express this by saying that there are (at least) 2 w 2 /9 differently labelled 2to-embeddings of K w,w,w for w ≡ 0 (mod 3). Since the two K 3,3,3 embeddings from [1] have the same black triangles, all the resulting K w,w,w embeddings may be taken to have the same black triangles. Furthermore, if n is odd (equivalently w ≡ 3 (mod 6)) then each of the K w,w,w embeddings may be taken to have a common (consistently oriented) parallel class of triangular faces in black.…”

“…or 2 m 2 /9 , and for m ≡ 1 (mod 6) we can take k = (m − 1)!. The case m = 3 is investigated in [1] where it is shown, inter-alia, that there are at least 2 n 2 /54−O(n) non-isomorphic 2to-embeddings of K n for n ≡ 7 or 19 (mod 36), and at least 2 2n 2 /81−O(n) for n ≡ 19 or 55 (mod 108). The latter estimate is achieved by a second application of the Construction, making use of aspects (1) and (2).…”

“…One of the former constructions can be viewed as a further generalisation of the construction given in [3] and [4]. All three enable us to produce 2 an 2 −o(n 2 ) non-isomorphic face 2-colourable triangula-tions of K n and K n,n,n in orientable and non-orientable surfaces for values of n lying in certain residue classes additional to the classes 7 and 19 modulo 36 covered in [1].…”

“…In a subsequent paper [1], the construction was generalised in a fashion which established the existence of 2 an 2 −o(n 2 ) (where a is a constant) non-isomorphic face 2-colourable triangulations of K n in orientable and non-orientable surfaces for certain residue classes, in particular n ≡ 7 or 19 (mod 36).…”