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Kevin Ryde
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NAME

Graph::Maker::Keller - create Keller graphs

SYNOPSIS

 use Graph::Maker::Keller;
 $graph = Graph::Maker->new ('Keller', N => 2);

DESCRIPTION

Graph::Maker::Keller creates a Graph.pm graph of a Keller graph. This is a graph form of Keller's conjecture on N-dimensional tilings by unit hypercubes.

Keller graph N has 4^N vertices numbered 0 to 4^N-1. These vertices are treated as base-4 integers. Edges are between vertices differing in two or more digit positions and at least one of which is difference 2 mod 4.

    N         0  1   2     3      4       5
    vertices  1, 4, 16,   64,   256,   1024, ...  4^N
    degree    0, 0,  5,   34,   171,    776, ...  4^N - 3^N - N
    edges     0, 0, 40, 1088, 21888, 397312, ... (4^N-3^N-N)*4^N / 2

Each vertex has the same degree since digit differences can all be taken mod 4. Each vertex degree is 4^N - 3^N - N. This is edges to 4^N vertices (including self) except no edge to those differing by only 0,1,3 in any or all digit positions, which is 3^N combinations (including no differences at all for vertex itself), and also no edge to vertices differing by 2 mod 4 at a single digit, which is N possible digit positions.

N=0 and N=1 have no edges since there are only 0 digits and 1 digit respectively, so nothing differs in 2 digit positions.

N=2 (16 vertices and 40 edges) is the Clebsh graph and the 16-cyclotomic graph.

Subgraph

subgraph => 1 gives the subgraph of the Keller graph induced by neighbours of vertex 0. This means the neighbours of vertex 0 (not including 0 itself) and the edges among those neighbours.

    subgraph => 1
    N         0  1   2     3      4       5
    vertices  0, 0,  5,   34,   171,    776, ... 4^N - 3^N - N
    edges     0, 0,  0,  261,  9435, 225990, ...

The number of vertices is the degree above. This subgraph is of interest since the Keller graph is vertex transitive so the problem of finding a clique of size k in the Keller graph is reduced to finding a clique of size k-1 in this subgraph. The size of the maximum clique is related to Keller's conjecture.

N=2 subgraph has no edges. Its vertices are 12, 21, 22, 23, 32 in base-4 and any pair of them either differ in only 1 digit or differ in 2 digits but without a difference 2 mod 4.

FUNCTIONS

$graph = Graph::Maker->new('Keller', key => value, ...)

The key/value parameters are

    N           => integer
    subgraph    => 0 or 1, default 0
    graph_maker => subr(key=>value) constructor, default Graph->new

Other parameters are passed to the constructor, either graph_maker or Graph->new().

If the graph is directed (the default) then edges are added both forward and backward between vertices. Option undirected => 1 creates an undirected graph and for it there is a single edge between vertices.

HOUSE OF GRAPHS

House of Graphs entries for graphs here include

N=2, https://hog.grinvin.org/ViewGraphInfo.action?id=975 (Clebsch)
N=3 subgraph, https://hog.grinvin.org/ViewGraphInfo.action?id=22730
N=4 subgraph, https://hog.grinvin.org/ViewGraphInfo.action?id=22732

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to these graphs include

    A202604    clique numbers (size of maximum clique)

SEE ALSO

Graph::Maker

O. H. Keller, "Uber die luckenlose Einfullung des Raumes mit Wurfeln", Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 163, 1930, pages 231-248.

LICENSE

Copyright 2015, 2016, 2017 Kevin Ryde

This file is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

This file is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with This file. If not, see http://www.gnu.org/licenses/.