K2‐Hamiltonian graphs: II

• Published in: Journal of graph theory Vol. 105; no. 4; pp. 580 - 611
• Main Authors: Goedgebeur, Jan, Renders, Jarne, Wiener, Gábor, Zamfirescu, Carol T.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.04.2024
• Subjects: Apexes; dot product; exhaustive generation; Graph theory; Graphs; hamiltonian cycle; hypohamiltonian; snark; Software
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23057

In this paper, we use theoretical and computational tools to continue our investigation of K2 ${K}_{2}$‐hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with K1 ${K}_{1}$‐hamiltonian graphs, that is, graphs in which every vertex‐deleted subgraph is hamiltonian. Perhaps surprisingly, there exist graphs that are both K1 ${K}_{1}$‐ and K2 ${K}_{2}$‐hamiltonian, yet non‐hamiltonian, for example, the Petersen graph. Grünbaum conjectured that every planar K1 ${K}_{1}$‐hamiltonian graph must itself be hamiltonian; Thomassen disproved this conjecture. Here we show that even planar graphs that are both K1 ${K}_{1}$‐ and K2 ${K}_{2}$‐hamiltonian need not be hamiltonian, and that the number of such graphs grows at least exponentially. Motivated by results of Aldred, McKay, and Wormald, we determine for every integer n $n$ that is not 14 or 17 whether there exists a K2 ${K}_{2}$‐hypohamiltonian, that is non‐hamiltonian and K2 ${K}_{2}$‐hamiltonian, graph of order n $n$, and characterise all orders for which such cubic graphs and such snarks exist. We also describe the smallest cubic planar graph which is K2 ${K}_{2}$‐hypohamiltonian, as well as the smallest planar K2 ${K}_{2}$‐hypohamiltonian graph of girth 5. We conclude with open problems and by correcting two inaccuracies from the first article.