New upper bounds on the order of cages

Let k≥2 and g≥3 be integers. A (k,g)-graph is a k-regular graph with girth (length of a smallest cycle) exactly g. A (k,g)-cage is a (k,g)-graph of minimum order. Let v(k,g) be the order of a (k,g)-cage. The problem of determining v(k,g) is unsolved for most pairs (k,g) and is extremely hard in the general case. It is easy to establish the following lower bounds for v(k,g): v(k,g)≥ k(k−1) (g−1)/2−2 k−2 for g odd, and v(k,g)≥ 2(k−1) g/2−2 k−2 for g even. The best known upper bounds are roughly the squares of the lower bounds. In this paper we establish general upper bounds on v(k,g) which are roughly the 3/2 power of the lower bounds, and we provide explicit constructions for such (k,g)-graphs. Mathematical Reviews Subject Numbers: 05C35, 05C38. Secondary: 05D99.