Locally pancyclic graphs
It was shown by \textit{J. A. Bondy} [Stud. Sci. Math. Hung. 4, 473-475 (1969; Zbl 0184.27702)] that if \(G\) is a graph of order \(n\) in which \(d_G(x) + d_G(y) \geq n\) for each pair of nonadjacent vertices \(x\) and \(y\) of \(G\), then \(G\) is either pancyclic or the complete bipartite graph \(K_{n/2,n/2}\). This result is generalized by proving a local version of the Bondy result. Let \(W\) be a set of at least \(3\) vertices of \(G\). It is shown that if \(d_G(x) + d_G(y) \geq n\) for each pair of nonadjacent vertices \(x\) and \(y\) of \(W\), then either for each \(i\) with \(3 \leq i \leq | W| \) there will be a cycle of \(G\) containing precisely \(i\) vertices of \(W\), or \(W = K_{2,2}\), or \(| W| = n\), and \(G = K_{n/2,n/2}\).
scientific article; zbMATH DE number 1346886