On the Panconnectivity of Graphs with Large Degrees and Neighborhood Unions

Abstract.

 Let G be a 2-connected graph on n vertices and δ be the minimum degree of G(V,E). Set NC=min{|N(u)∪N(v)|: u,v∈V, uv∉E}. In this paper, we show that if for any independent set {u,v,w} of G, we have min{|N(u)∪N(v)|+d(w), |N(u)∪N(w)|+d(v), |N(v)∪N(w)|+d(u)}≥n+1, then for any integer 7≤i≤n and any vertices x,y such that {x,y} is not a cut vertex set of G, there exists a path connecting u,v with i vertices in G. In particular, if G is 3-connected, then G is [7,n]-panconnected. Using this result, we can prove that if NC+δ≥n+1, then G is vertex pancyclic. This was conjectured by Faudree, Gould, Jacobson and Lesniak.