A couple of questions related to edge-disjoint cycles.

Let $K_n = (V,E)$ be the complete graph on $|V|=n$ nodes. Two cycles are 'edge disjoint' if they do not share any edges.

What is the size of the largest collection of edge-disjoint cycles of length 3 in Kn?

Consider a spanning tree sub-graph of $K_n$ (so any spanning tree on $n$ nodes). Is there an efficient algorithm for adding $c$ edges such that each edge creates a cycle of length 3 and is edge-disjoint with all other cycles added?

Any references or ideas would be greatly appreciated!

thanks...