Hamiltonicity of 3-connected line graphs

Weihua Yang, Liming Xiong, Hongjian Lai, Xiaofeng Guo
2012 Applied Mathematics Letters  
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
more » ... line graph of a graph with at most 9 vertices of degree 3 is Hamiltonian; examples show that all conditions are sharp.
doi:10.1016/j.aml.2012.02.032 fatcat:vnp5xljyjbaqffhgvlgy7pzi3y