Packing triangles in a graph and its complement

How few edge-disjoint triangles can there be in a graph G on n vertices and in its complement ? This question was posed by P. Erdos, who noticed that if G is a disjoint union of two complete graphs of order n/2 then this number is n(2)/12 + o(n(2)). Erdos conjectured that any other graph with n vertices together with its complement should also contain at least that many edge-disjoint triangles. In this paper, we show how to use a fractional relaxation of the above problem to prove that for every graph G of order n, the total number of edge-disjoint triangles contained in G and 1 is at least n(2)/13 for all sufficiently large n. This bound improves some earlier results. We discuss a few related questions as well. (C) 2004 Wiley Periodicals, Inc.