Chvatal-Erdos Condition for Hamiltonicity in Digraphs

Our purpose is to determine the minimum integer \(f_i(m)\) (\(g_i(m)\), \(h_i(m)\) respectively) for every natural \(m\), such that every digraph \(D\), \(f_i(m)\)-connected, (\(g_i(m)\), \(h_i(m)\)-connected respectively) and \(\alpha^i(D) \leq m\) is hamiltonian (D has a hamilton path, D is hamilton connected respectively), (\(i = 0,1, 2\)). We give exact values of \(f_i(m)\) and \(g_i(m)\) for some particular values of \(m\). We show the existence of \(h_2(m)\) and that \(h_2(1) = 1\), \(h_2(2) = 4\) hold.