I designed a 2-player board game "Schneider von Gent" where either
the graph Schneider-17 or the Gent-17 are used as game board.
Click!
Frank Schneider found an alternating planar graph with
30 vertices, 33 faces, and 61 edges. This refutes a
conjecture of Karl Scherer, who thought that the difference
between number of vertices and faces coul differ by atmost 2.
Nicolas van Cleemput (Gent University) proved with computer help
that there
do not exist alternating planar graphs with 18 vertices.
A planar graph is called alternating, when the
following conditions are fulfilled:
* There are no neighbouring vertices with the same degree.
* There are no neighbouring faces with the same number
of sides. (Attention: the outer face of the graph is
also a face!)
* Each vertex has degree at least 3.
* Each face has at least 3 sides.
My motivation to introduce alternating planar graphs in 2008 came from
Karl Scherer's nice alternating tilings. Alternating
tilings are tilings with
several (at least two) types of tiles where no two tiles of the same
type touch each other. See for instance Karl's beautiful 20- and
21-solutions for squaring the square
here.
The 21-solution became the base for a variant of my game
EinStein würfelt nicht. Karl Scherer also has provided a nice
website on alternating planar graphs in the Wolfram Demonstrations
Project, click
here.
Back in 2008, I was in search of alternating planar graphs where
one of the following criteria is met:
* Number of vertices is very small -
or
* Number of vertices plus number of faces
is smaller than 34 (34 = 17 + 17 was from the record holder
Schneider-17, by Frank Schneider).
Observe that in the Schneider graph all degrees (for vertices and faces) are from 3, 4, 5. Furthermore, this graph is self-dual. Jan Kristian Haugland (alias "Tasmanian Devil" on LittleGolem) found that in each alternating planar graph with the 3,4,5-restriction the number of vertices and the number of faces are equal! His proof may be found here (somewhere in the middle of the lengthy thread).
For some time it was an open question whether each alternating planar graph with (3,4,5)-restriction on the degrees is self-dual. Frank Schneider gave a clear "No" by showing that several of "his" (3,4,5)-apg's are not self-dual.