Thomassen's conjecture implies polynomiality of 1-Hamilton-connectedness in line graphs

A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every x ∈ V (G), and G is 2-edge-Hamilton-connected if the graph G + X has a hamiltonian cycle containing all edges of X for any X ⊂ E+(G) = {xy| x, y ∈ V (G)} with 1 ≤ |X| ≤ 2. We prove that Thomassen’s conjecture (every 4-connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4-connected line graph is 1-Hamilton-connected and/or 2-edgeHamilton-connected. As a corollary, we obtain that Thomassen’s conjecture implies polynomiality of both 1-Hamilton-connectedness and 2-edge-Hamilton-connectedness in line graphs. Consequently, proving that 1-Hamilton-connectedness is NP-complete in line graphs would disprove Thomassen’s conjecture, unless P=NP.