Second Hamiltonian Cycles in Claw-Free Graphs

• Main Authors: Hossein Esfandiari, Colton Magnant, Pouria Salehi Nowbandegani, Shirdareh Haghighi
• Format: Article
• Language: English
• Published: Georgia Southern University 2015-01-01
• Series: Theory and Applications of Graphs
• Subjects: claw-free graph; Sheehan’s conjecture; second Hamiltonian cycle
• Online Access: https://digitalcommons.georgiasouthern.edu/tag/vol2/iss1/2
• ISSN: 2470-9859

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.