On the Circumferences of Regular 2-Connected Graphs

LetGbe a 2-connectedd-regular graph onn⩽rd(r⩾3) vertices andc(G) denote the circumference ofG. Bondy conjectured thatc(G)⩾2n/(r−1) ifnis large enough. In this paper, we show thatc(G)⩾2n/(r−1)+2(r−3)/(r−1) for any integerr⩾3. In particular,Gis hamiltonian ifr=3. This generalizes a result of Jackson. Examples to show that the bond forc(G) is sharp and that Bondyʹs conjecture does not hold ifris allowed to take non-integer values are given.