Hamiltonian degree conditions for tough graphs (Q1896350)

Let \(G\) be a graph with degree sequence \(d_1, d_2,\dots, d_n\) and \(t\) a positive integer. It is shown that if \(G\) is \(t\)-tough and for any \(i\) with \(t\leq i< n/2\) we have \(d_i\leq i\Rightarrow d_{n- i+ 1}\geq n- i\), then \(G\) is hamiltonian. This result generalizes a theorem of Chvátal for the case \(t= 1\). A similar sufficient condition is obtained for pancyclicity of graphs.