New graphs related to (p,6) and (p,8)-cages

Alain Bretto (1) , Alain Faisant (2) , Luc Gillibert (3)

1 Equipe CODAG - Laboratoire GREYC - UMR6072
2 LA MUSE - Laboratoire de Mathématiques Unifiées de Saint-Etienne
3 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

Constructing regular graphs with a given girth, a given degree and the fewest possible vertices is hard. This problem is called the cage graph problem and has some links with the error code theory. G-graphs can be used in many applications: symmetric and emisymmetric graph constructions, (Bretto and Gillibert (2008) [12]), hamiltonicity of Cayley graphs, and so on. In this paper, we show that G-graphs can be a good tool to construct some upper bounds for the cage problem. For p odd prime we construct (p, 6)-graphs which are G-graphs with orders 2p2 and 2p2 − 2, when the Sauer bound is equal to 4(p − 1)3. We construct also (p, 8)-G-graphs with orders 2p3 and 2p3 −2p, while the Sauer upper bound is equal to 4(p − 1)5.

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

Alain Bretto, Alain Faisant, Luc Gillibert. New graphs related to (p,6) and (p,8)-cages. Computers & Mathematics with Applications, 2011, 62, pp.2472-2479. ⟨hal-00947639⟩