The odd graph O_n of order n is a graph having vertices given by the (n - 1)-subsets of {1, ..., 2n - 1} 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 n-subsets of {1, ..., 2n + 1} is sometimes also used, leading to a shifting of the index by one (e.g., West 2000, Ex. 1.1.28, p. 17).
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