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 G? This question was posed by P. Erdó́s, who noticed that if G is a disjoint union of two complete graphs of order n=2 then this number is n=12 þ o(n). Erdó́s 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 G is at least n=13 for all sufficiently large n. This bound improves some earlier results. We discuss a few related questions as well. 2004 Wiley Periodicals, Inc. J Graph Theory 47: 203–216, 2004