Upper Bounds on the Order of Cages

  • F. Lazebnik
  • V. A. Ustimenko
  • A. J. Woldar

Abstract

Let k2 and g3 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(k1)(g1)/22k2 for g odd, and v(k,g)2(k1)g/22k2 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.

Published
1996-11-21