A short proof of Fanʹs theorem

In this note, we give a new short proof of the following theorem: Let G be a 2-connected graph of order n. If for any two vertices uu and vv with d(u,v)=2,d(u,v)=2,max{d(u),d(v)}⩾c/2,max{d(u),d(v)}⩾c/2, then the circumference of G is at least c, where 3⩽c⩽n3⩽c⩽n and d(u,v)d(u,v) is the distance between uu and vv in G.