A Construction of a 3/2‐Tough Plane Triangulation With No 2‐Factor

ABSTRACT In 1956, Tutte proved the celebrated theorem that every 4‐connected planar graph is Hamiltonian. This result implies that every more than ‐tough planar graph on at least three vertices is Hamiltonian and so has a 2‐factor. Owens in 1999 constructed non‐Hamiltonian maximal planar graphs of toughness arbitrarily close to and asked whether there exists a maximal non‐Hamiltonian planar graph of toughness exactly . In fact, the graphs Owens constructed do not even contain a 2‐factor. Thus the toughness of exactly is the only case left in asking the existence of 2‐factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by constructing a maximal ‐tough plane graph with no 2‐factor, answering the question asked by Owens as well as by Bauer, Broersma, and Schmeichel.