A shortness exponent for r‐regular r‐connected graphs

Abstract

Abstract Let r ≧ 3 be an integer. It is shown that there exists ε= ε( r ), 0 < ε < 1, and an integer N = N(r ) > 0 such that for all n ≧ N (if r is even) or for all even n ≧ N (if r is odd), there is an r ‐connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than n ε vertices.