A criterion for some Hamiltonian graphs to be Hamilton-connected

Let M = {G / Kn,n ~ G ~ Kn V for some n 2:: 3} where V is the join operation. The author and N.K. Khachatrian proved that a connected graph G of order at least 3 is Hamiltonian if d(u) + d(v) 2:: IN(u) U N(v) U N(w)1 for each triple of vertices U,V,w with d(u,v) = 2 and w E N(u) n N(v) (where N( x) is the neighborhood of x). Here we prove that a graph G satisfying the above conditions is Hamilton-connected if and only if G is 3-connected and G r:J. M.