Degree conditions for weakly geodesic pancyclic graphs and their exceptions

Abstract

Let $G$ be a graph on $n$ vertices. Let $\sigma _2(G)=\text {min}\{d_G(u)+d_G(v):u,v\in {V(G)};uv\notin {E(G)}\}$ when $G$ is not complete, otherwise set $\sigma _2(G)=\infty$. A graph $G$ is said to be weakly geodesic pancyclic if for each pair of vertices $u,v\in {V(G)}$, every shortest $u,v$-path lies on a cycle of length $k$ where $k$ is an integer between the length of a shortest cycle containing the $u,v$-path and $n$. In this paper, we will show that if $\sigma _2(G)\ge {n+1}$ then $G$ is either weakly geodesic pancyclic or belongs to one of four exceptional classes of graphs, which are completely determined. Our results generalize some recent results of Chan et al. (Discret Appl Math 155:1971-1978, 2007).