Two types of path structures of graphs

Abstract

In 2005, Rahman and Kaykobad proved that if G is a 2-connected graph with n vertices and d(u) + d(v ) + (u,v)⩾n+1 for each pair of distinct non-adjacent vertices u,v in G, then G is traceable [Information Processing Letters, 94 (2005), 1, 37–41]. In this present paper, we prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+(u,v)⩾n for each pair of distinct non-adjacent vertices u,v in G, then G is traceable or G belongs to a class of wellstructured exceptional graphs