TGT2020th International Workshop on Topological Graph

TheoryYokohama National University

November 27th, 2008

Generalized Polyhedral Suspensions

Serge Lawrencenko

労 Rou (in Japanese)

Lao (in Chinese)

Akina Nakamori’s song “Tears Aren't a Decoration” (“Kazarijanainoyo Namidawa”)

Polyhedral suspensions find an application in art and design. They effectively represent tears.

Песня Акины Накамори “Слезы не украшение” (“Казаридзянайнойо Намидава”)

Многогранные подвески находят применение в искусстве и дизайне. Они эффектно изображают слезы.

Definition of a Geometric Polyhedral Suspension

A geometric polyhedral suspension is a 2-dimensional polyhedron C with triangular faces in Euclidean 3-space, satisfying the following two conditions:

(1) All but two of the vertices of C lie in one plane called the equatorial plane.(2) The two exceptional nonequatorial vertices are nonadjacent in the graph of C.

The simplicial complex determined by the equatorial vertices, edges, and triangles of C is called the equator of C. The two exceptional vertices are located in different half spaces and are called the north pole N and south pole S.

1983: The first example of toroidal polyhedralsuspension:Toroidal Bipyramidal

Hexadecahedron, T ——>

S. L., “All irreducible triangula-tions of the torus are realizable in E^3 as polyhedra”

Second Prize in the 1983 Student Research Paper Com-petition conducted annually by the Dept. of Math. & Mech., Moscow State University

A bipyramid is a polyhedral suspension in which the poles both adjacent to each vertex in the equator.

Construction for building the toroidal bipyramid with 8 vertices:

N and S are placed in front and behind the equatorial plane. Adjoin the triangles determined by N and the edges of the blue cycle as well as the triangles determined by S and the edges of the red cycle.

This is toroidal bipyramidal hexadecahedron T.

Theorem 1: There exists a regular 2-dimensional polyhedron with 8 vertices in Euclidean 4-space. That polyhedron is a toroidal regular hexadacahedron.

Proof:

● T is a regular toroidal triangulation with 8 vertices.

● Graph G(T) = K_{2,2,2,2} = 1-skeleton of the 4D-octahedron.

● T itself is a subcomplex of the 2-skeleton of the 4D-octahedron. ■

We propose a new invariant:

s(K) — the spatiality of a simplicial 2-complex K is

the minimum number of pairwise nonadjacent vertices whose removal from K leaves a simplicial complex planar.

If the carrier of K is homeomorphic to the sphere, then s(K) = 1.

If it is homeomorphic to a closed surface other than the sphere, s(K) ≥ 2.

If it is homeomorphic to a closed nonorientable surface,s(K) ≥ 3.

Combinatorial Definition of Abstract Polyhedral Nonspherical Suspension:

An abstract polyhedral nonspherical suspension of given genus g (g ≠ 0) is an abstract simplicial 2-complex C having Euler characteristic χ(C) = 2 – 2g and spatiality s(C) = 2, and satisfying the following two conditions:

(1) The link of each vertex v in C is a Hamilton cycle through the neighbors of v.

(2) Every 1-simplex of C is incident with precisely two 2-simplexes of C. ■

This definition can serve as a combinatorial algorithm for testing whether a given abstract 2-complex K is a polyhedral suspension of given genus g. By a theorem of Gross and Rosen, and Mohar, s(K) is indeed a combinatorial invariant.

Theorem 2:For each positive integer g, there exists a bipyramid, gT,of genus g.

● gT is a triangulation of the closed orientable surface of genus g.

● gT is the connected sum of g tori T.

Equator of 2T, the double-torus bipyramid:

.

Equator of 3T, the triple-torus bipyramid:

Corollary 1: There exist thickness-two graphs of arbitrarily large genus.

Proof: The genus(G(gT)) = g. The thickness(G(gT)) = 2 for any g ≥ 1

because

G(gT) = G(equator(gT)) U K_{2, n–2} is the union of two planar graphs.

Here n = |V(G(gT))| is the number of vertices of gT, and the complete bipartite graph K_{2, n–2} is determined by this 2-partition of the vertex set: V(G(gT)) = {N,S} U V(G(eqtr(T))).■

Corollary 2: There exist planar graphs of arbitrarily large closed 2-cell maximum genus.

Proof: The equatorial graph E = G(equator(gT)) is planar for any g and admits a cellular embedding in Σ_g.

=> 0 = genus (E) ≤ g ≤ max genus (E).

We therefore rediscover Ringeisen’s Theorem (1973)that states the existence of planar graphs of arbitrarily large maximum genus.

Furthermore, we have reinforced it as stated. ■

A family of graphs is said to have linear crossing number if there is a constant c such that

cr(G) ≤ c|V(G)| for any graph in the family.

● Pach and Tóth (2006): toroidal graphs with bounded degree have linear crossing number.

● Hliněný and Salazar (2007): a polynomial-time algorithm for estimating the crossing number

of toroidal graphs with bounded degree.

Corollary 3: The family of bipyramidal graphs {G(gT)} has linear crossing number with c ≤ 2.

Proof: G(gT) can be drawn in the plane with 10g crossings.

cr(G(gT)) ≤ 10g = 2n–4 ≤ 2n,(g ≥ 2)

where n = the number of vertices of gT. ■

Clearly, cr(G) ≥ genus(G).

In an attempt to bring together cr(G) and genus(G),one can modify the above construction, judiciously removing some nonequatorial edges, to reach a graph still having genus = g but with crossing number less than 10g.

Conjecture. For each g ≥ 2, there is a subgraph of G(gT) with genus and crossing number both equal to g.