Generation and new infinite families of K2-hypohamiltonian graphs

We present an algorithm which can generate all pairwise non-isomorphic K2-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for K2-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which K2hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite K2-hypohamiltonian graphs. Furthermore, we describe a new operation for creating K2-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar K2-hypohamiltonian graph of order n for every integer n >= 134. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence K2-hypohamiltonian graphs with large maximum degree and size.