Graphs with Maximal Even Girth

Abstract

In this paper we examine the classGof simple undirected, connected graphs of diameterd> 1, girth2d,and for anygɛG, if a pair of nodes are at distancedfrom each other, then that pair of nodes is connected bytdistinct paths of lengthd, t> 1. (The girth ofgis the length of the smallest circuit ing.)We establish, in § 2, that for allgɛG, gis regular.We establish necessary conditions for the existence of elements ofG.IfgɛG,we adopt the notationg=g(d, t, v, n),wherevis the valence ofgandnis the number of nodes. It is of course possible forg, hɛG, g ≠ h,and for givend, t, v, nto have bothg(d, t, v, n)andh(d, t, v, n).In particular, we show that ifd= 2,t≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular giventvalue.