The odd graph 
of order 
is a graph having vertices given by the 
-subsets of 
such that two vertices are connected by an edge
iff the associated subsets are disjoint (Biggs 1993, Ex.
8f, p. 58). Some care is needed since the convention of defining the odd graph
based on the 
-subsets
of 
is sometimes also used,
leading to a shifting of the index by one (e.g., West 2000, Ex. 1.1.28, p. 17).
By the definition of the odd graph using using the prevalent convention, the number of nodes in 
is 
, where 
is a binomial coefficient.
For 
, 2, ..., the first few values are
1, 3, 10, 35, 126, ... (OEIS A001700).
is isomorphic to the singleton
graph, 
to the triangle graph 
, and 
to the Petersen graph
(Skiena 1990, p. 162). The Kneser graph 
is a generalization of the odd
graph, with 
corresponding to 
.
The bipartite Kneser graph is a generalization
of the bipartite double graph of the odd
graph, with 
corresponding to 
(which, like 
,
is distance-transitive; Brouwer et
al. 1989, p. 222).
is regular
of vertex degree 
and has graph diameter 
(Biggs 1976). The girth
of 
is 6 for 
(West 2000, p. 17; adjusting the indexing convention
to the more common definition based on 
subsets).
The odd graphs are distance-transitive, and therefore also distance-regular. They
are also automorphic graphs (Biggs 1976). It
is conjectured that 
is of class 1 except for the cases 
and 
a power of two (Fiorini and Wilson 1977).
Balaban (1972) exhibited Hamiltonian cycles for 
and 5, Meredith and Lloyd (1972, 1973)
found cycles for 
and 7, and Mather (1976) showed a Hamiltonian cycle
for 
(Shields and Savage).
Since the odd graph is a special case of the Kneser graph, its independence number follows from the value
for 
as
 |
Odd graphs are implemented in the Wolfram Language as FromEntity[Entity["Graph",
"Odd", n]],
and precomputed properties for small odd graphs are implemented as GraphData[
"Odd", n
].
to 
."
In Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972).
Southend: Inst. Math. Appl., pp. 229-236, 1972.Meredith, G. H. J.
and Lloyd, E. K. "The Footballers of Croam." J. Combin. Theory
Ser. B 15, 161-166, 1973.Mütze, T. "On Hamilton
Cycles in Graphs Defined by Intersecting Set Systems." Not. Amer. Soc. 74,
583-592, 2024.Shields, I. and Savage, C. D. "A Note on Hamilton
Cycles in Kneser Graphs." 
."
Exercise 1.1.28 in