A graph is said to be H^*-connected if it is either Hamilton-connected or Hamilton-laceable. S. Wagon conjecture that all connected vertex-transitive graphs are H^*-connected with the following exceptions: cycle graphs, the dodecahedral graph, Petersen graph, Coxeter graph, triangle-replaced Petersen graph, and triangle-replaced Coxeter graph. The conjecture can be restated as, "Every Hamiltonian vertex-transitive graph is H^*-connected except cycle graphs C_n for n>=5 and the dodecahedral graph." It was been verified up to n = 31 nodes.

Hamilton-connected graph | Hamilton-laceable graph | nonhamiltonian vertex-transitive graph | triangle-replaced graph