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Hamiltonicity of 3-connected line graphs
Authors:Weihua Yang  Liming Xiong  Hongjian Lai  Xiaofeng Guo
Institution:1. LRI, C.N.R.S.-Université de Paris-sud, 91405-Orsay cedex, France;2. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China;3. Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA;4. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, PR China;5. School of Mathematical Science, Xiamen University, Xiamen Fujian 361005, PR China;6. Department of Mathematics, Jiangxi Normal University, PR China;7. Department of Mathematics, Qinghai University for Nationalities, PR China
Abstract:Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. conjectured H. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571–576] that every 3-connected, essentially 4-connected line graph is Hamiltonian. In this note, we first show that the conjecture posed by Lai et al. is not true and there is an infinite family of counterexamples; we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is Hamiltonian; examples show that all conditions are sharp.
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