Odd Graph

The odd graph , for a positive integer , is a simple graph having vertices given by the -subsets of such that two distinct vertices are connected by a graph 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 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 . For , the bipartite double graph of is the bipartite Kneser graph , which, like , is distance-transitive (Brouwer et al. 1989, p. 222).

For , is regular of vertex degree and has graph diameter (Biggs 1976). The singleton graph has vertex degree and graph diameter zero. 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). For , it is conjectured that is of class 1 except for the cases and a power of 2 (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).

For , a Hamilton decomposition (or quasi-Hamilton decomposition) of is equivalent to the existence of pairwise edge-disjoint Hamiltonian cycles (with the remaining edges forming a perfect matching when is odd). Since is the non-Hamiltonian Petersen graph, it is not a Hamilton decomposable graph. Meredith and Lloyd (1973) showed that is the edge-disjoint union of two Hamiltonian cycles and a perfect matching and that is the edge-disjoint union of three Hamiltonian cycles. Together with the explicit certificates stored in GraphData for and , this gives Hamilton decompositions or quasi-Hamilton decompositions for with , 5, 6, and 7.

It is conjectured that contains pairwise edge-disjoint Hamiltonian cycles for every (Biggs 1979). This is precisely the assertion that all such odd graphs are Hamilton decomposable under the extended convention. The theorem that every with is Hamiltonian (Mütze et al. 2021) establishes only one Hamiltonian cycle and does not settle this decomposition conjecture.

For , the independence number follows from that of the Kneser graph as

The singleton graph has independence number 1.

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].