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

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 semi-symmetric 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 2p^2 and 2p^2-2, when the Sauer bound is equal to 4(p-1)^3. We construct also (p,8)-G-graphs with orders 2p^3 and 2p^3-2p, while the Sauer upper bound is equal to 4(p-1)^5.

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