OFFSET
1,1
COMMENTS
There is an obvious bijection between the cycles of the graph of a polyhedron and the partitions of its faces (or vertices of the dual polyhedron) into two connected subsets; see second table in Example section. Also, there is a bijection between those partitions and the minimal edge cuts of the graph of the dual polyhedron. - Pontus von Brömssen, Sep 29 2025
LINKS
Seiichi Manyama, Python program (github)
Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Eric Weisstein's World of Mathematics, Cubical Graph
Eric Weisstein's World of Mathematics, Octahedral Graph
Eric Weisstein's World of Mathematics, Dodecahedral Graph
Eric Weisstein's World of Mathematics, Icosahedral Graph
EXAMPLE
graph \ n-cycle | 3 4 5 6 7 8 9 10 11 12 13 ...
-------------------+-------------------------------------------------
tetrahedral graph | 4 3
cubical graph | 0 6 0 16 0 6
octahedral graph | 8 15 24 16
dodecahedral graph | 0 0 12 0 0 30 20 36 120 100 60 ...
icosahedral graph | 20 30 72 240 720 1620 2680 3336 2880 1280
.
Numbers of partitions of the faces into 2 connected parts, by size of the smallest part:
graph \ size | 1 2 3 4 5 6 7 8 9 10
-------------------+-----------------------------------------
tetrahedral graph | 4 3
cubical graph | 6 12 10
octahedral graph | 8 12 24 19
dodecahedral graph | 12 30 80 210 480 356
icosahedral graph | 20 30 60 140 312 690 1420 2730 4560 2916
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Seiichi Manyama, Dec 10 2022
STATUS
approved