Biregular Cages of Odd Girth
Geoffrey Exoo, Robert Jajcay · Journal of Graph Theory · 2015
Biregular -cages are graphs of girth g that contain vertices of degrees r and m and are of the smallest order among all such graphs. We show that for every and every odd , there exists an integer m0 such that for every even , the biregular -cage is of order equal to a natural lower bound analogous to the well-known Moore bound. In addition, when r is odd, the restriction on the parity of m can be removed, and there exists an integer m0 such that a biregular -cage of order equal to this lower bound exists for all . This is in stark contrast to the result classifying all cages of degree k and girth g whose order is equal to the Moore bound.