Independent Dominating Sets and a Second Hamiltonian Cycle in Regular Graphs

AbstractIn 1975, John Sheehan conjectured that every Hamiltonian 4-regular graph has a second Hamiltonian cycle. Combined with earlier results this would imply that every Hamiltonianr-regular graph (r⩾3) has a second Hamiltonian cycle. We shall verify this forr⩾300

Second Hamiltonian Cycles in Claw-Free Graphs

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs

Second Hamiltonian Cycles in Claw-Free Graphs

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs