Two Sufficient Conditions for Hamilton and Dominating Cycles

We prove that ifG is a 2-connect graph of size q the number of edges andminimumdegree δwith δ ≥ 2q/3 /12−1/2, where 11 when δ 2 and 31 when δ ≥ 3, then each longest cycle in G is a dominating cycle. The exact analog of this theorem for Hamilton cycles follows easily from two known results according to Dirac and Nash-Williams: each graph with δ ≥ q 5/4 − 1/2 is hamiltonian. Both results are sharp in all respects.