On Hamiltonian Regular Graphs

In this paper we shall determine, when 1 = 6, bounds for numbers f(k, I) and F{k, 1) defined as follows: f{k, l)/F(k, I) is defined to be the smallest integer n for which there exists a regular graph/Hamiltonian regular graph of valency k and girth I having n vertices. The problem of determining minimal regular graphs of given girth was first considered by Tutte [9]. Bounds for f(k, I) have been obtained by Erdos and Sachs [2], while certain values of F(K, 6) have been found by Karteszi [6]. We shall determine an improved upper bound for f(k, 6) and also an upper bound for F(k, 6); our results will be best possible, in an asymptotic sense. In our constructions we shall utilize elementary properties of finite projective planes, and properties of the distribution of primes. The author wishes to acknowledge the assistance of the following of his colleagues during the course of this research: Professors W. McWorter, Z. A. Melzak, R. Westwick, and G. K. White; he is indebted to the referee for suggested simplifications in §6.