Edgeless Go

    Keywords: Variant

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Using some terminology from mathematical graph theory, a Go variant is a member of the Edgeless Go family of variants when:

  1. The rules of play are the same as normal Go
  2. All points in the finite graph G upon which plays are made are linked to precisely four other points.

The second condition just concerns the graph upon which the game is played. So the term “edgeless Go” is also used for graphs.

In mathematical terminology, the second condition just says the board is a finite 4-regular graph (see the [ext] wikipedia article on regular graphs).

Locally grid Go

A proper subclass of Edgeless Go variants is here called “locally grid Go”; it is obtained by strengthening the second condition for Edgeless Go above by requiring the graph G to be “locally like the grid ⊞”, or “locally grid”, according to the precise definition given in the online mathematical article [ext] “locally grid graphs”. An interesting result in the paper is a classification of all graphs that are locally grid. It turns out that all locally grid graphs are embeddable in the torus or the Klein bottle!

Examples and counterexamples

Most of the following examples have been proposed by various people on various forums and social media.

  • Standard Toroidal Go (‘t-Go’) is an example of locally grid Go.
  • Rectangular t-Go: as for t-Go, but on an m x n grid with m ≠ n, is an example of locally grid Go.
  • Twisted t-Go: t-Go with a twist. This is another example of locally grid Go. The following illustration is taken from the article cited above ([ext] “locally grid graphs”).

a twisted toroidal graph
left: a (7,5) toroidal graph with a twist of 2. Right: an illustration of a “twisted torus” in 3-D space

  • Cylindrical Go: instead of removing all four edges from normal Go, only remove two. Strictly speaking this is not an example of Edgeless Go, as points at the ends of the cylinder have three liberties, not four.
  • standard n x n graph embedded in the Klein surface

Klein surface
A Klein surface

This is an example of locally grid Go, but unlike for toroids and twisted toroids, the graph is not homogenous (more precisely, the graph is not “[ext] vertex transitive”).

  • Projective plane

Projective plane
A projective plane

The standard n x n graph embedded in a projective plane provides an example of a graph that is not “edgeless Go”. This can be see in the following representation of the 3 x 3 projective graph - three lines of “wraparound” are added.

i h g c b a i h g
f e d f e d f e d
c b a i h g c b a
g h i a b c g h i
d e f d e f d e f
a b c g h i a b c
i h g c b a i h g
f e d f e d f e d
c b a i h g c b a

The point “a” only has three neighbours: “b”, “d” and “i” (twice!).

  • Chamfered cube

Chamfered cube
A chamfered cube

This provides another example of edgeless Go where the graph is not locally-grid like.

  • Round Go: another example of edgeless Go where the graph is not locally-grid like.

Edgeless Go last edited by Malcolm on January 3, 2020 - 16:42
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