Smallest maximally nonhamiltonian graphs

Smallest maximally nonhamiltonian graphs II

“…Isaacs [6] was the first to construct an infinite family {Jk} of such graphs. Clark and Entringer [5] have shown that the Jk and variations of them are maximally nonhamiltonian graphs which implies f(n) = 3n/2 for all even n > 36 and f(n) = (3n + i)/2 or (3n + 3)/2 for all odd n > 55. In [3] it was shown that additional variations of the Jk are maximally nonhamiltonian graphs and, hence, f(n)= (3n + 1)/2 for n -5 (rood 8) with n > 21.…”

“…Clark and Entringer [5] have shown that the Jk and variations of them are maximally nonhamiltonian graphs which implies f(n) = 3n/2 for all even n > 36 and f(n) = (3n + i)/2 or (3n + 3)/2 for all odd n > 55. In [3] it was shown that additional variations of the Jk are maximally nonhamiltonian graphs and, hence, f(n)= (3n + 1)/2 for n -5 (rood 8) with n > 21. We show that further variations of the Jk are maximally nonharniltonian graphs and, hence, f(n) ---(3n + 1)/2 for all odd n > 53.…”

“…We show that further variations of the Jk are maximally nonharniltonian graphs and, hence, f(n) ---(3n + 1)/2 for all odd n > 53. As in [3], a computer has been used in settling some small cases. Our notation and terminology generally follows Bollob~is [1].…”

Maximally non-hamiltonian graphs of girth 7

“…It is easy to see that the Isaacs' snarks Jk (Fig. 7), which are examples of MNH graphs [1] for odd k > 7, have girth 6. For many years the Coxeter graph (Fig.…”

The saturation function of complete partite graphs

“…The value of sat(n, C m ) is unknown (even asymptotically) for any other fixed m; various bounds are proved in [1,17]. Also, the aggregate outcome of papers [6,19,11,10,12,32], with the final gaps filled by computer search, determines sat(n, C n ), the Hamilton cycle case, for all n ≥ 3.…”