Vertex pancyclicity and new sufficient conditions

Abstract

For a graph G, δ(G) denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x) + d(y) ≥ n for each pair of non-adjacent vertices x, y in G ,t henG is pancyclic or G = Kn/2,n/2. In 2001, Xu proved that, if G is a 2-connected graph of order n ≥ 6a nd|N (x) ∪ N (y) |+ δ(G) ≥ n for each pair of non-adjacent vertices x, y in G ,t henG is pancyclic or G = Kn/2,n/2. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n ≥ 6a nd |N (x) ∪ N (y) |+ d(w) ≥ n for any three vertices x, y ,w of d(x, y) = 2a ndwx or wyE(G) in G ,t henG is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results. We generalize two well-known degree sum and neighborhood union for characterizing Hamiltonian graphs, in particular for vertex-pancyclic. We first give a few definitions and some notation. We consider only finite undirected graphs with no loops or multiples. We denote by δ(G) the minimum degree of G, and Kh the complete graph of order h .I fu is a vertex and H is a subgraph of G, then let NH (u) ={ v ∈ V (H ) : uv ∈ E(G)} be the vertex set of H that are adjacent to vertex u, and if S is a vertex set or subgraph of G, then set NH (S) =∪ u∈S NH (u) and set N (u )= N (u) ∪{ u} .L etG − H and G(S) denote the subgraphs of G induced by V (G) − V (H ) and S, respectively. If Cm = x1x2 ··· xm x1 is a cycle of order m ,l etN + Cm (u) ={ xi+1 : xi ∈ NCm (u)}, N − Cm (u) = {xi−1 : xi ∈ NCm (u)}, and N ± Cm (u) = N +