Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs
Roman Kužel, Zdeněk Ryjáček, Petr Vrána · Journal of Graph Theory · 2011
Abstract 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