Smallest maximally nonhamiltonian graphs

Abstract

A graph G is maximally nonhamiltonian iff G is not hamiltonian but G + e is hamiltonian for each edge e in Gc, i.e., any two non-adjacent vertices of G are ends of a hamiltonian path. Bollobás posed the problem of finding the least number of edges, f(n), possible in a maximally nonhamiltonian graph of order n. Results of Bondy show that f(n) ≤3/2n for n ≤ 7. We exhibit graphs of even order n ≥ 36 for which the bound is attained. These graphs are the "snarks", Jk, of Isaacs and mild variations of them. For odd n ≥ 55 we construct graphs from the graphs Jk showing that in this case, f(n) = 3 n + 1/2 or 3 n + 3/2 and leave the determination of which is correct as an open problem. Finally we note that the graphs Jk, k ≤ 7 are hypohamiltonian cubics with girth 6. © 1983 Akadémiai Kiadó.