TGT20 - PowerPoint PPT Presentation

About This Presentation
Title:

TGT20

Description:

Construction for building the ... Adjoin the triangles determined by N and the edges of the blue cycle as well as the triangles determined ... PowerPoint Presentation ... – PowerPoint PPT presentation

Number of Views:38
Avg rating:3.0/5.0
Slides: 18
Provided by: lawren127
Category:

less

Transcript and Presenter's Notes

Title: TGT20


1
TGT20 20th International Workshop on Topological
Graph Theory Yokohama National University November
27th, 2008 Generalized Polyhedral
Suspensions Serge Lawrencenko ? Rou
(in Japanese) Lao (in
Chinese)
2
Akina Nakamoris song Tears Aren't a Decoration

(Kazarijanainoyo Namidawa) Polyhedra
l suspensions find an application in art and
design. They effectively represent tears.
3
????? ????? ???????? ????? ?? ?????????

(???????????????? ????????)??????????
?? ???????? ??????? ?????????? ? ????????? ?
???????. ??? ???????? ?????????? ?????.
4
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.
5
1983 The first example of toroidal
polyhedral suspension Toroidal
Bipyramidal Hexadecahedron, T gt S. L.,
All irreducible triangula- tions of the torus
are realizable in E3 as polyhedra Second
Prize in the 1983 Student Research Paper
Com- petition conducted annually by the Dept. of
Math. Mech., Moscow State University

6
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.
7
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.
8
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.
9
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 ?(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.
10
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.
11
Equator of 2T, the double-torus bipyramid
12
Equator of 3T, the triple-torus bipyramid
.
13
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, n2 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, n2 is determined by this
2-partition of the vertex set V(G(gT)) N,S U
V(G(eqtr(T))).
14
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 S_g. gt
0 genus (E) g max genus
(E). We therefore rediscover Ringeisens Theorem
(1973) that states the existence of planar graphs
of arbitrarily large maximum genus.
Furthermore, we have reinforced it as stated.
15
A family of graphs is said to have linear
crossing number if there is a constant c such
that cr(G) cV(G) for any graph in the
family. ? Pach and Tóth (2006) toroidal graphs
with bounded degree have linear crossing number.
? Hlinený and Salazar (2007) a polynomial-time
algorithm for estimating the crossing number of
toroidal graphs with bounded degree.
16
  • 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 2n4 2n, (g 2)
  • where n the number of vertices of gT.

17
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.
Write a Comment
User Comments (0)
About PowerShow.com