**Abstract:** Two *n*-vertex graphs *G*_{1} and *G*_{2} *pack* if *G*_{1} and *G*_{2} can be expressed as edge-disjoint subgraphs of *K*_{n}. Special cases of the problem of whether two given graphs pack include problems on existence of fixed subgraphs, on forbidden subgraphs, and on equitable coloring. Graph packing has also been applied in studying computational complexity of graph properties. Let *\Delta(G)* denote the maximum degree of a vertex in *G*.

Bollobás and Eldridge (also Catlin, independently) conjectured that if *|V(G*_{1})|=|V(G_{2})|=n and *(\Delta(G*_{1})+1)(\Delta(G_{2})+1) <= n+1, then *G*_{1} and *G*_{2} pack. If true, this conjecture would be sharp, and it would be a considerable extension of the Hájnal-Szemerédi theorem on equitable coloring. The conjecture has only been proved in cases where *G*_{1} is highly degenerate, or *\Delta*_{1} <= 2, or *\Delta*_{1}=3 and *n* is huge.

In this talk, we take a different approach to the conjecture. Given *n*-vertex graphs *G*_{1} and *G*_{2}, we prove that if *\Delta(G*_{1}), \Delta(G_{2}) >= 400 and *(\Delta(G*_{1})+1)(\Delta(G_{2})+1) <= 0.6n+1, then *G*_{1} and *G*_{2} pack. This is joint work with H. Kaul and A. Kostochka.