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.
