A generalized Keller graph has as its vertices the 
-tuples over the integers 0, 1, ..., 
. Two vertices are adjacent if they differ by exactly 
in at least one coordinate and differ in at least two coordinates (Brakensiek et
al. 2022). By definition, it is a Cayley graph
of the additive group 
whose connection set
consists of the vectors having at least one coordinate equal to 
and at least two nonzero coordinates. This set is inverse-closed
because negation modulo 
preserves both properties.
The generalized Keller graph with parameters 
and 
is denoted 
, or 
after the notation 
of Łysakowska (2023). Its vertex set therefore
has 
elements. The ordinary 
-dimensional Keller graph 
is the special case 
.
Special cases, including the natural boundary cases with 
or 
, are summarized in the following table.
 | graph |
 |  -empty graph |
 | ladder rung graph  |
 | Clebsch graph |
 |  rook graph |
 |  -dimensional Keller graph |
For 
,
generalized Keller graphs are vertex-transitive
and regular, with vertex
degree 
.
Their chromatic number is 
, and their independence
number is 
for 
.
In addition, Łysakowska (2023) proved that all generalized Keller graphs are
Hamiltonian and class
1.
Corrádi and Szabó (1990) introduced the graphs 
to give a convenient finite graph-theoretic formulation
of Keller's conjecture, translating the search
for face-sharing-free periodic cube tilings into a clique problem. A clique in 
has size at most 
, and one attaining this bound gives a face-sharing-free
tiling of 
-dimensional
space. It therefore disproves Keller's conjecture
in dimension 
and all higher dimensions. Łysakowska (2023) later used the term generalized
Keller graph for the family and studied its graph-theoretic properties.
For the remaining seven-dimensional case, Brakensiek et al. (2022) encoded the clique search as a satisfiability problem
and used symmetry breaking to prove that none of 
, 
, and 
contains a clique of size 128. This settled Keller's
conjecture by proving it in dimension seven.
Generalized Keller graphs will be implemented in a future version of the Wolfram Language as GraphData[
"GeneralizedKeller",
n,
s
].
