A characterization of panconnected graphs satisfying a local ore-type condition

A. S. Asratian DEPARTMENT OF MATHEMATICS UNIVERSITY OF UMEA S-907 87, UMEA, SWEDEN DEPARTMENT OF MATHEMATICAL CYBERNETICS YEREVAN STATE UNIVERSITY YEREVAN, 375049, REPUBLIC OF ARMENIA R. Haggkvist DEPARTMENT OF MATHEMATICS UNIVERSITY OF UMEA S-907 87, UMEA, SWEDEN G. V. Sarkisian DEPARTMENT OF MATHEMATICAL CYBERNETICS YEREVAN STATE UNIVERSITY YEREVAN, 375049, REPUBLIC OF ARMENIA It is well known that a graph G of order p 2: 3 is Hamilton-connected if d(u) +d(v) 2: p+ 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) 2: IN(u) u N(v) u N(w)I + 1 for any path uwv with uv (/. E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) s k s IV(G)I 1, there is an x y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k E {3, 4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of 96 JOURNAL OF GRAPH THEORY G belongs to a triangle and a quadrangle. 84 sons, lnc. Our results imply some results of Williamson, Faudree, and Schelp.