Radius, girth and minimum degree

Abstract

AbstractThe objective of the present paper is to study the maximum radius of a connected graph of order , minimum degree and girth at least . Erdős, Pach, Pollack and Tuza proved that if , that is, the graph is triangle‐free, then , and noted that up to the value of the additive constant, this upper bound is tight. In this paper we shall determine the exact maximum. For larger values of little is known. We settle the order of the maximum for and 12, and prove an upper bound for every even , which we conjecture to be tight up to a constant factor. Finally, we show that our conjecture implies the so‐called Erdős girth conjecture.