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Cage (graph theory) — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Cage_(graph_theory)
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-06T17:34:12.245Z
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Cage (graph theory) — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Cage_(graph_theory) (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_bc2ff37a072b (Metraipolis archive dc_bc2ff37a072b, sha256 1a91227c532f123f6d268f180adf25177da20c3bbdf06d4aff15e4439946f72e)

BibTeX

@misc{dcbc2ff37a072b,
  title = {Cage (graph theory) — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Cage_(graph_theory)},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_bc2ff37a072b; sha256 1a91227c532f123f6d268f180adf25177da20c3bbdf06d4aff15e4439946f72e},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Cage_(graph_theory). Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In the mathematical field of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r, g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has a length of exactly g. An (r, g)-cage is an (r, g)-graph with the smallest possible number of vertices, among all (r, g)-graphs. A (3, g)-cage is often called a g-cage. It is known that an (r, g)-graph exists for any combination of r ≥ 2 and g ≥ 3. It follows that all (r, g)-cages exist. If a Moore graph exists with degree r and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth g must have at least 1 + r ∑ i = 0 ( g − 3 ) / 2 ( r − 1 ) i {\displaystyle 1+r\sum _{i=0}^{(g-3)/2}(r-1)^{i}} vertices, and any cage with even girth g must have at least 2 ∑ i = 0 ( g − 2 ) / 2 ( r − 1 ) i {\displaystyle 2\sum _{i=0}^{(g-2)/2}(r-1)^{i}} vertices. Any (r, g)-graph with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of r and g. For instance there are three non-isomorphic (3, 10)-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3, 11)-cage: the Balaban 11-cage (with 112 vertices).

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «graph theory»).
Fonte: Wikipedia in inglese, voce «Cage (graph theory)» — https://en.wikipedia.org/wiki/Cage_(graph_theory)
Wikidata Q841007 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q841007
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Cage_(graph_theory)?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1340267028, consultata il 2026-10-06T16:01:04.207Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 840f7533c26e3c0dcbb16dd3f6a006e306b082ea5e42fce848bb83f62961473e