$K_2$-Hamiltonian Graphs: I

Abstract

Motivated by a conjecture of Grünbaum and a problem of Katona, Kostochka, Pach, and Stechkin, both dealing with non-hamiltonian n-vertex graphs and their (n − 2)-cycles, we investigate K 2 -hamiltonian graphs, i.e. graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph. In this first part, we prove structural properties, and show that there exist infinitely many cubic non-hamiltonian K 2 -hamiltonian graphs, both of the 3-edge-colourable and the non-3-edgecolourable variety. In fact, cubic K 2 -hamiltonian graphs with chromatic index 4 (such as Petersen's graph) are a subset of the critical snarks. On the other hand, it is proven that non-hamiltonian K 2 -hamiltonian graphs of any maximum degree exist. Several operations conserving K 2 -hamiltonicity are described, one of which leads to the result that there exists an infinite family of non-hamiltonian K 2 -hamiltonian graphs in which, asymptotically, a quarter of vertices has the property that removing such a vertex yields a non-hamiltonian graph. We extend a celebrated result of Tutte by showing that every planar K 2 -hamiltonian graph with minimum degree at least 4 is hamiltonian. Finally, we investigate K 2 -traceable graphs, and discuss open problems.

K2‐Hamiltonian graphs: II

“…. By an operation of the last author [41], this statement is also true for K 2 -hypohamiltonian graphs. Grötschel asked whether there is a bipartite graph admitting no hamiltonian path, but in which every vertex-deleted subgraph does contain a hamiltonian path, see [18, problem 4.56].…”

“…In [41], lemma 3 is used in the proof of theorem 1, whose statement remains valid. The only change that must be made in the proof of theorem 1 is that in its last paragraph, the sentence "By construction, Γ k contains a cubic vertex y k such that no vertex of N y ( ) k is exceptional."…”

“…Thus, the answer to the aforementioned question is that there exists a hypohamiltonian graph of order n if and only if ∈ n {10, 13, 15, 16} or ≥ n 18 is an integer. The first author determined that all (seven) hypohamiltonian graphs on at most 16 vertices are K 2 -hamiltonian, while the last author showed that the generalised Petersen graph GP (2,11) is K 2 -hamiltonian [41] (that GP (2,11) is hypohamiltonian was shown by Bondy [4]). Combining these graphs via Lemmas 3 and 4, we obtain hypohamiltonian K 2 -hamiltonian graphs of order 10, 13, 15, 16, 18, and every integer which is at least 20; a detailed account is given in the proof of Theorem 3.…”

“…Finally, in Section 4, we conclude the paper with an erratum concerning [41] and discuss potential future directions of research.…”

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

“…Grünbaum conjectured that there exists no $$n$$‐vertex graph of circumference $$n-2$$ such that any pair of vertices is avoided by a longest cycle [13]. Both remain open, but an important special case of the question of Katona, Kostochka, Pach, and Stechkin is being studied in [40]. We do know that there exist planar cubic 3‐connected graphs in which any pair of vertices is avoided by a longest cycle [25].…”