Notes on vertex pancyclicity of graphs

Abstract

In (2012) [7], Kewen Zhao and Yue Lin introduced a new sufficient condition for pancyclic graphs and proved that if G is a 2-connected graph of order n⩾6 with |N(x)∪N(y)|+d(w)⩾n for any three vertices x,y,w of d(x,y)=2 and wx or wy∉E(G), then G is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result generalized the two results of Bondy in 1971 and Xu in 2001. In this paper, we first prove that if G is a 2-connected graph of order n⩾6 with |N(x)∪N(y)|+d(w)⩾n for any three vertices x,y,w of d(x,y)=2 and wx or wy∉E(G), then each vertex u of G with d(u)⩾3 is 5-pancyclic or G=Kn/2,n/2, and we also show that our result is best possible. On the basis of this result, we prove that there exist at least two pancyclic vertices in G or G=Kn/2,n/2. In addition, we give a new proof of a result in Cai (1984) [2].