On the Hamiltonian index

Abstract For simple connected graphs that are neither paths nor cycles, we define h ( G ) = min{ m : L m ( G ) is Hamiltonian} and l ( G ) = max{ m : G has an arc of length m that is not both of length 2 and in a K 3 }, where an arc in G is a path in G whose internal vertices have degree two in G . We prove that h ( G )⩽ l ( G ) + 1. As consequences, we obtain theorems of Chartrand and Wall and of Lesniak-Foster and Williamson. We also characterize those graphs that satisfy l ( G ) + 1 = h ( G ). This characterization provides counterexamples to a previous result in [5].