On the hamiltonicity of a planar graph and its vertex‐deleted subgraphs

Abstract

Tutte proved that every planar 4-connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex-deleted subgraphs are hamiltonian. We here prove that if in a planar n-vertex graph with minimum degree at least 4 at least n − 5 vertex-deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every c < 1 there exists a nonhamiltonian polyhedral n-vertex graph with minimum degree at least 4 containing cn hamiltonian vertex-deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex-deleted subgraphs as well as Bondy's meta-conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex-deleted subgraphs are traceable, must itself be traceable.

On non-hamiltonian polyhedra without cubic vertices and their vertex-deleted subgraphs

“…We briefly comment on the observation used in Figure 2 to obtain a triangulation-for results on the interaction between n-and (n − 1)-cycles in n-vertex triangulations we refer to [15]. For all graphs proving the statement of Theorem 3, we chose the graph J to be, for every replacement, the octahedron minus the edges of a triangle (as before, the identified vertices shall be the three 2-valent vertices of the octahedron minus the edges of a triangle).…”

“…It was recently shown that in the statement of ( †), the second "n" can be replaced by n − 5, see [15]. It is natural to wonder how much more this threshold can be lowered.…”

“…Note that the exact threshold in the generalisation of ( †) that we are discussing here remains unknown: if ζ is the largest integer such that every planar graph whose minimum degree is at least 4 and in which n − ζ vertex-deleted subgraphs contain a hamiltonian cycle, must itself contain a hamiltonian cycle, then we only know that 5 ≤ ζ ≤ 23. We do have an asymptotic solution to this problem, i.e., for every c < 1 there is a polyhedral n-vertex graph without vertices of degree 3 in which at least cn vertex-deleted subgraphs contain a hamiltonian cycle, but which itself contains no hamiltonian cycle [15].…”