Panconnectedness and Pancyclicity in Graphs

with Local Ore-Type Conditions II A.S. Asratian, R H aggkvist and G.V. Sarkisian University of Ume a January, 1994 Abstract. It is well known due Ore that a graph G of order at least 3 is Hamiltonconnected if d(u) + d(v) jV (G)j + 1 for each pair of nonadjacent vertices u and v. We consider here connected graphs G of order at least 3 where d(u) + d(v) jN(u) [N(v) [N(w)j+ 1 for each triple of vertices u; v; w with d(u; v) = 2 and w 2 N(u) \N(v) where N(x) denote the neighborhood of a vertex x. We prove that a graph G with this condition has the following properties: a) For each pair of nonadjacent vertices x; y of G and for each integer n, d(x; y) n jV (G)j 1 there is an x y path of length n. b) For each edge xy of G and for each integer n (excepting maybe one n 2 f3; 4g) there is a cycle of length n containing xy. It implies that G is panconnected (and also edge pancyclic) if and only if each edge of G lies on a triangle and on a quadrangle.