Edge-Diameter of a Graph and Its Longest Cycles

Given a graph G and X , Y ⊂ V ( G ) , d G ( X , Y ) is the distance between X and Y and the edge diameter d i a m e ( G ) is the greatest distance between two edges of G . In this note, we consider edge diameter of a graph and its longest cycles and prove the following: Let G be a connected graph other than a tree with d i a m e ( G ) ≤ d ′ , then G has a longest cycle D such that d G ( e , D ) ≤ d ′ - 1 for any edge e of G , furthermore, if G is 2-connected, then d G ( e , C ) ≤ d ′ - 1 for any longest cycle C and any edge e of G . Let H be a 3-connected simple graph with d i a m e ( H ) ≥ d ′ . Then H has a cycle of length at least 2 d ′ + 3 if H is not K 4 , furthermore, H has a cycle of length at least 2 d ′ + 4 if d ′ ≥ 4 .