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].