A generalized Keller graph has as its vertices the n-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 taking additive inverses 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 hypercube 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 are implemented in the Wolfram Language as GraphData[
"GeneralizedKeller",
n,
s
].
