Hamiltonicity of 3connected line graphs 
 
Authors:  Weihua Yang Liming Xiong Hongjian Lai Xiaofeng Guo 
 
Institution:  1. LRI, C.N.R.S.Université de Parissud, 91405Orsay 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 4connected line graph is Hamiltonian. Lai et al. conjectured H. Lai, Y. Shao, H. Wu, J. Zhou, Every 3connected, essentially 11connected line graph is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571–576] that every 3connected, essentially 4connected 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 3connected, essentially 4connected 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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