Longest cycles in r-regular r-connected graphs

Let r ≥ 3 be an integer, and ε > 0 a real number. It is shown that there is an integer N(r, ε) 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 εn vertices. That is, the number ε > 0, no matter how small, is a “shortness coefficient” for the family of r-valent regular r-connected graphs.