The minimum size and maximum diameter of an edge-pancyclic graph of a given order

A k-cycle in a graph is a cycle of length k. A graph G of order n is called edge-pancyclic if for every integer k with 3≤k≤n, every edge of G lies in a k-cycle. We give lower and upper bounds on the minimum size of a simple edge-pancyclic graph of a given order, and determine the maximum diameter of such a graph. In the 3-connected case, the precise minimum size is determined. We also determine the minimum size of a graph of a given order with connectivity conditions in which every edge lies in a triangle.