Notes on vertex pancyclicity of graphs

• We consider the vertex-pancyclicity of a class of graph G.
• We prove that each vertex u of G   with d(u)>2d(u)>2 is 5-pancyclic or G is a complete bipartite graph.
• We prove that there exist at least two pancyclic vertices in G or G is a complete bipartite graph.
• We give a new proof of a result in Cai (1984) [2].

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⩾6n⩾6 with |N(x)∪N(y)|+d(w)⩾n|N(x)∪N(y)|+d(w)⩾n for any three vertices x,y,wx,y,w of d(x,y)=2d(x,y)=2 and wx   or wy∉E(G)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⩾6n⩾6 with |N(x)∪N(y)|+d(w)⩾n|N(x)∪N(y)|+d(w)⩾n for any three vertices x,y,wx,y,w of d(x,y)=2d(x,y)=2 and wx   or wy∉E(G)wy∉E(G), then each vertex u of G   with d(u)⩾3d(u)⩾3 is 5-pancyclic or G=Kn/2,n/2G=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/2G=Kn/2,n/2. In addition, we give a new proof of a result in Cai (1984) [2].