Odd

Summary

An odd cycle is a cycle whose length is odd. The odd girth of a non-bipartite graph is the length of its shortest odd cycle. An odd hole is a special case of an odd cycle: one that is induced and has four or more vertices.

An odd vertex is a vertex whose degree is odd. By the handshaking lemma every finite undirected graph has an even number of odd vertices.

An odd ear is a simple path or simple cycle with an odd number of edges, used in odd ear decompositions of factor-critical graphs; see ear.

An odd chord is an edge connecting two vertices that are an odd distance apart in an even cycle. Odd chords are used to define strongly chordal graphs.

An odd graph is a special case of a Kneser graph, having one vertex for each $(n - 1)$-element subset of a $(2n - 1)$-element set, and an edge connecting two subsets when their corresponding sets are disjoint.