Collapsible Graphs and Hamiltonicity of Line Graphs

NSFC [11171279]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 D (3)(G) the set of vertices of degree 3 of graph G. For , define d(e) = d(u) + d(v) - 2 the edge degree of e, and . Denote by lambda (m) (G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with , and is collapsible; a 3-edge-connected simple graph with , and is collapsible; a 3-edge-connected graph with , , and with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with , and with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with , and 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 is Hamiltonian