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Graph-Maker-Other-19
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/ Graph::Maker::Keller

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.

The 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 2 or more digit positions, 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 (so a regular graph) 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 no digits and a single digit respectively, so nothing differs in 2 digit positions.

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

The graph diameter is 2, for N>=2. A pair of vertices u,v not connected by an edge either don't have a digit differing 2 mod 4, or do but that is the sole difference. For the latter, a common intermediate is formed simply by a different digit at one of the other positions. For the former, take an intermediate differing from u by 2 mod 4 at one digit position, and differing from v by 2 mod 4 at another digit position.

Subgraph

subgraph => 1 gives the subgraph of the Keller graph induced by neighbours of vertex 0. This means the neighbours of vertex 0 (and not 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 and 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

1310     N=0  (singleton)
52       N=1  (4 singletons)
975      N=2  (Clebsch, 16-cyclotomic)
34165    N=3

And subgraph option

62       N=2  (5 singletons)
22730    N=3
22732    N=4

OEIS

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

  A284838    num edges
  A202604    clique numbers (size of maximum clique)
  A258935    independence number
  A292056    Wiener index
  A301571    num vertices at distance 2 (3^N + N - 1)

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.

The second DIMACS challenge 1993 included the N=4 subgraph among its benchmarks for comparing clique-finding algorithms, http://dimacs.rutgers.edu/Challenges/

HOME PAGE

http://user42.tuxfamily.org/graph-maker-other/index.html

LICENSE

Copyright 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/.