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 $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$ of a simple edge-pancyclic graph of order $n.$ We give lower and upper bounds on $f(n),$ and determine the maximum diameter of such a graph. In the $3$-connected case, the precise value of $f(n)$ 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.