Odd girth

Definition

The odd girth of an undirected graph is defined as the minimum of the lengths of cycles of odd length (i.e., subgraphs that are isomorphic to cycle graphs). The odd girth is taken to be $\infty$ if there is no cycle of odd length.

Particular cases

Value of odd girth	What it tells us about the graph
$\infty$	There are no cycles of odd length. Equivalently, the graph is a bipartite graph.
strictly greater than 3, i.e., at least 5	The graph is a triangle-free graph.

Related invariants

• Girth refers to the minimum of the lengths of all cycles, whether even or odd.
• Even girth refers to the minimum of the lengths of all cycles of even length.