The authors consider connected graphs \(G\) such that, for each triple \(\{u, v, w\}\) such that \(uwv\) is a path of length 2 in \(G\) and \(uv\not\in E(G)\), we have \(d(u)+ d(v)\geq |N(u)\cup N(v)\cup N(w)|+ 1\). The degree of a vertex \(u\) is denoted by \(d(u)\) and the neighborhood by \(N(u)\). It is shown that such graphs are panconnected and are edge pancyclic (except for possibly a missing cycle of length 3 or 4). This generalizes a result of the reviewer and R. H. Schelp [Acta Math. Acad. Sci. Hungar. 25, 313-319 (1974; Zbl 0294.05119)].