OEIS A000066: Cage graphs

An exploration of the cubic (3-regular) cage graphs: the smallest number of vertices needed for a graph to have a given girth. We discuss what A000066 counts, known results for girths up to 12, the remarkable unique cubic cage with 112 vertices and girth 11 proved by McKay and Mirvold, and how higher-degree cages (such as A0006856 for 4-regular cages) expand the landscape. We’ll also touch on practical connections to network design and coding theory, and why the OEIS serves as a living map of these interlinked ideas.

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