A note on the upper bound and girth pair of (k;g) ( k ; g ) -cages
A (k;g) ( k ; g ) -cage  is a k k -regular graph of girth g g with minimum order. In this work, for all k?3 k ? 3 and g?5 g ? 5 odd, we present an upper bound of the order of a (k;g+1) ( k ; g + 1 ) -cage in terms of the order of a (k;g) ( k ; g ) -cage, improving a previous result by Sauer of 1967. We also show that every (k;11) ( k ; 11 ) -cage with k?6 k ? 6 contains a cycle of length 12, supporting a conjecture by Harary and Kovcs of 1983.
girth pair cage kronecker product