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‐edge‐Hamilton‐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 . © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012