Chvatal-Erdos Condition for Hamiltonicity in Digraphs

Abstract

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 ${\alpha }^i(D)\le m$ 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.