Advertisement
Published in:
01-02-2016
Degree conditions for weakly geodesic pancyclic graphs and their exceptions
Published in: Journal of Combinatorial Optimization | Issue 2/2016
Log inActivate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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).