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

  • Authors:

  • Roman Kužel;Zdeněk Ryjáček;Petr Vrána

  • Affiliations:

  • Department of Mathematics, University of West Bohemia and Institute for Theoretical Computer Science (ITI), Charles University, P.O. Box 314 306 14, Pilsen, Czech Republic;-;-

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

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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, © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Czech Ministry of Education; Contract grant numbers: 1M0545 and MSM 4977751301 (to R. K. and Z. R.))