The rook graph (confusingly called
the grid by Brouwer et al. 1989,
p. 440) and also sometimes known as a lattice graph (e.g., Brouwer) is the graph Cartesian product of complete graphs,
which is equivalent to the line graph of the complete
bipartite graph .
This is the definition adopted for example by Brualdi and Ryser (1991, p. 153),
although restricted to the case . This definition corresponds to the connectivity graph of
a rook chess piece (which can move any number of spaces in a straight line-either
horizontally or vertically, but not diagonally) on an chessboard.

Define an
Latin square graph as a graph whose vertices are the elements of the Latin square
and such that two vertices being are if they lie in the same row or column or contain
the same symbol. These graphs correspond to the rook graph and the minimum
vertex colorings of the
rook graph give the distinct Latin squares.

Aubert and Schneider (1982) showed that rook graphs admit Hamiltonian decomposition, meaning they are class 1 when they have even vertex
count and class 2 when they have odd vertex count
(because they are odd regular).

Aubert, J. and Schneider, B. "Décomposition de la somme cartésienne d'un cycle et de l'union de deux cycles hamiltoniens
en cycles hamiltoniens." Disc. Math.38, 7-16, 1982.Brouwer,
A. E. "Lattice Graphs." http://www.win.tue.nl/~aeb/drg/graphs/Hamming.html.Brouwer,
A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular
Graphs. New York: Springer-Verlag, 1989.Brouwer, A. E.
and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries."
In Enumeration
and Design: Papers from the conference on combinatorics held at the University of
Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson
and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122,
1984.Brualdi, R. and Ryser, H. J. §6.2.4 in Combinatorial
Matrix Theory. New York: Cambridge University Press, p. 152, 1991.Godsil,
C. and Royle, G. "Latin Square Graphs." §10.4 Algebraic
Graph Theory. New York: Springer-Verlag, pp. 226-230, 2001.Karavaev,
A. M. "FlowProblem: Statistics of Simple Cycles." http://flowproblem.ru/paths/statistics-of-simple-cycles.Perepechko,
S. N. and Voropaev, A. N. "The Number of Fixed Length Cycles in an
Undirected Graph. Explicit Formulae in Case of Small Lengths."van
Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their
Spectrum?" Lin. Algebra Appl.373, 139-162, 2003.