Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and vice versa.

A graph is bipartite if and only if it does not contain an odd cycle.

Correspondingly, the chromatic number of an odd cycle is three.

Surely he would break out of this odd cycle if he climbed up another level.

More generally, a graph is bipartite if and only if it has no odd cycles (Kőnig, 1936).

Removing this edge from the odd cycle leaves a path, which may be colored using the two colors for its subgraph.

This result, known today as the Erdős-Pósa theorem, cannot be extended to odd cycles.

For 2D layouts double patterning compliance errors occur when there are odd cycles of minimum spaces.

However, if there exists at least one odd cycle, then no 2-edge-coloring is possible.

Brooks' theorem states that with two exceptions (cliques and odd cycles) at most Δ colors are needed.