Hamilton cycles in almost-regular 2-connected graphs (Q757425)

Let \(k\) and \(s\) be integers, \(1\leq s\leq 4\). Let \(G\) be a graph whose vertices have degrees between \(k\) and \(k+s\), and \(| G| \leq 3k-c(s),\) \(1\leq s\leq 3\), or \(| G| \leq 2.5k-c(s),\) \(s=4\) 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 \(n=| G| \leq 3k-1\) then \(G\) is hamiltonian unless \(n\) is odd and \(\alpha (G)=1/2(n+1).\)