Hamilton Cycles in Almost-Regular 2-Connected Graphs

Let k and s be integers, 1 ≤ s ≤ 4. Let G be a graph whose vertices have degrees between k and k + s, and |G| ≤ 3k − c(s), 1 ≤ or |G| ≤ 2.5k − c(s), s = 4, for suitable constants c(s) depending on s. We obtain a necessary and sufficient condition for G to be hamiltonian. In particular we show that if s = 1 and n = |G| ≤ 3k − 1 then G is hamiltonian unless n is odd and α(G) = 12(n + 1).