On upper bounds of odd girth cages

Abstract

We give a construction of k -regular graphs of girth g using only geometrical and combinatorial properties that appear in any ( k ; g + 1 ) -cage, a minimal k -regular graph of girth g + 1 . In this construction, g ≥ 5 and k ≥ 3 are odd integers, in particular when k − 1 is a power of 2 and ( g + 1 ) ∈ { 6 , 8 , 12 } we use the structure of generalized polygons. With this construction we obtain upper bounds for the ( k ; g ) -cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 5 , 7 , 11 .