Zigzag and central circuit structure of two-faced plane graphs

author: Michel-Marie Deza, Ecole Normale Superieure

Description

A zigzag in a k-valent plane graph G is a circuit of edges, such that any two, but not three consecutive edges belong to the same face. A railroad in G is a circuit of evengonal faces, such that any face is adjacent to its neighbors on opposite edges. Boundary circuits of a railroad are two ”parallel” zigzags if k = 3, or, in a 4-valent graph, two such central circuits. We consider the zigzag and railroad structure of two-faced 3- and 4-valent plane graphs (generalizations of Platonic polyhedra) and their connections with other problems.

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*Slides*
0:00	- Zigzags and central circuits for 3- or 4-valent plane graphs - Announcement
0:46	Zigzags and central circuits for 3- or 4-valent plane graphs
1:05	I. k-valent two-faced polyhedra
1:18	Polyhedra and planar graphs
1:26	Classes and their generation
2:44	Examples
3:03	Finite isometry groups - Page 1
3:08	k-connectedness - Page 1
3:36	k-connectedness - Page 2
3:39	k-connectedness - Page 3
3:56	Medial graph - Page 1
4:59	Medial graph - Page 2
5:35	Inverse medial graph - Page 1
5:57	Inverse medial graph - Page 2
6:20	Inverse medial graph - Page 3
6:25	- Inverse medial graph - Page 2 - Part 2
6:29	- Inverse medial graph - Page 3 - Part 2
6:38	II. Zigzags and central circuits
6:42	Central circuits - Page 1
6:48	Central circuits - Page 2
6:50	Central circuits - Page 3
6:52	Central circuits - Page 4
7:32	Central circuits - Page 5
7:45	Central circuits - Page 6
8:17	Zigzags - Page 1
8:26	Zigzags - Page 2
8:34	Zigzags - Page 3
8:52	Zigzags - Page 4
8:57	Zigzags - Page 5
9:10	Zigzags - Page 6
9:33	- Zigzags - Page 5 - Part 2
9:49	- Zigzags - Page 6 - Part 2
11:33	Intersection types for zigzags - Page 1
11:55	Intersection types for zigzags - Page 2
11:58	Intersection types for central circuits - Page 1
12:03	Intersection types for central circuits - Page 2
12:05	Intersection types for central circuits - Page 3
12:16	Intersection types for central circuits - Page 4
12:18	Intersection types for central circuits - Page 5
12:20	Duality and types
12:46	Medial, zigzags and central circuits - Page 1
12:57	Medial, zigzags and central circuits - Page 2
13:12	Notation
13:57	Zigzags versus spanning trees
15:08	Intersection of two simple ZC-circuits
15:33	Bipartite graphs
15:48	III. Railroad structure and tightness
15:59	Railroads, 4-valent case
18:41	Railroads, 3-valent case
19:10	First IPR fullerene with self-int. railroad
20:55	Triple self-intersection
21:37	Railroads with triple points in small 4n
22:21	Removing central circuits - Page 1
22:54	Removing central circuits - Page 2
23:01	Removing central circuits - Page 3
23:12	Removing zigzags - Page 1
23:14	Removing zigzags - Page 2
23:15	Removing zigzags - Page 3
23:17	Removing zigzags - Page 4
23:19	Extremal problem
24:48	Tight with simple central circuits - Page 1
25:37	Tight with simple central circuits - Page 2
25:49	Tight with simple zigzags
26:29	Tight 5n with simple zigzags - Page 1
28:35	Tight 5n with simple zigzags - Page 2
28:53	Tight Fn with only simple zigzags
29:47	IV. Goldberg-Coxeter construction
30:39	V. Parametrizing graphs
30:46	Parametrizing graphs qn
32:44	The structure of graphs 3n
32:49	z- and railroad-structure of graphs 3n
32:52	General theory
33:00	Number of parameters
33:31	Conjecture on 4n
33:34	More conjectures
33:35	VI. Zigzags on surfaces
33:38	Zigzags of 2-complexes (surface maps)
34:09	Zigzags of regular maps
34:41	Lins trialities
35:40	Example: Tetrahedron
36:21	Bipartite skeleton case
36:25	VII. Zigzags on n-dimensional complexes
36:29	Zigzags on n-dimensional polytopes
37:29	Zigzags of regular/semiregular polytopes
38:14	Zigzags of reg. and semireg. polyhedra - Part 1
38:22	Zigzags of reg. and semireg. polyhedra - Part 2
38:24	Regular-faced and Conway’s polytopes
38:28	VIII. Special fullerenes 5n
38:42	All 5n with hexagons in 1 ring
39:02	All 5n with hexagons in (> 1) rings
39:12	z-uniform 5n with n <= 60
40:09	Two 5-60 with z-vector
40:12	z-uniform IPR 5n with n <= 100
40:14	IPR z-knotted 5n with n <= 100
41:04	Perfect matching on 5n graphs
41:08	Statistics of z-knotted 5n with n <= 74
42:46	IV. Goldberg-Coxeter construction
42:48	The construction
43:25	Gluing the pieces
43:39	Final steps
44:00	Goldberg-Coxeter for Cube
44:46	- Questions

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