A graph is said to be -connected if it is either Hamilton-connected or Hamilton-laceable.
S. Wagon (pers. comm., May. 20, 2013; Dupuis and Wagon 2014) conjecture that all connected vertex-transitive graphs are -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 -connected except cycle graphs for and the dodecahedral graph." It was been verified up to nodes.