Generation and new infinite families of K2-hypohamiltonian graphs

Abstract

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 K2-hypohamiltonian 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.