Collapsible Graphs and Hamiltonicity of Line Graphs

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai (Combinatorics and Graph Theory, vol 95, World Scientific, Singapore, pp 53–69; Conjecture 8.6 of 1995) conjectured that every 3-edge connected and essentially 6-edge connected graph is collapsible. Denote D3(G) the set of vertices of degree 3 of graph G. For e = uv ∈ E(G), define d(e) = d(u)+ d(v)− 2 the edge degree of e, and ξ(G) = min{d(e) : e ∈ E(G)}. Denote by λm(G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with ξ(G) ≥ 7, and λ3(G) ≥ 7 is collapsible; a 3-edge-connected simple graph with ξ(G) ≥ 7, and λ3(G) ≥ 6 is collapsible; a 3-edge-connected graph with ξ(G) ≥ 6, λ2(G) ≥ 4, and λ3(G) ≥ 6 with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with ξ(G) ≥ 6, and λ3(G) ≥ 5 with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with ξ(G) ≥ 5, and λ2(G) ≥ 4 with at most 9 vertices of degree 3 is collapsible. As a corollary, we show that a 4-connected line graph L(G) with minimum degree at least 5 and |D3(G)| ≤ 9 is Hamiltonian. The research is supported by NSFC (No. 11171279). W. Yang (B) Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China e-mail: [email protected] W. Yang · H. Li Laboratoire de Recherche en Informatique, C.N.R.S., University de Paris-sud, 91405 Orsay cedex, France H.-J. Lai Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA X. Guo School of Mathematical Science, Xiamen University, Xiamen, Fujian 361005, China