# The Bollobás-Eldridge-Catlin Conjecture on graph packing

**Conjecture (BEC-conjecture)**If and are -vertex graphs and , then and pack.

A pair of -vertex graphs and are said to if they are edge-disjoint subgraphs of the complete graph on vertices.

The main conjecture in the area of graph packing is the abovementioned conjecture by Bollobás, Eldridge [BE] and Catlin [C].

In support of the BEC-conjecture, Sauer and Spencer [SS] proved that if and are -vertex graphs and then and pack.

Given a graph , denotes the line graph of and denotes the number . Kostochka and Yu [KY1] proved that if and are two -vertex graphs with , then and pack with the following exceptions: (1) is a perfect matching and is either with odd or contains or (2) is a perfect matching and is with odd or contains .

Kostachka and Yu [KY2] conjectured that if and are -vertex graphs with then and pack.

## Bibliography

*[BE] B. Bollabás and S. E. Eldridge, Maximal matchings in graphs with given maximal and minimal degrees, Congr. Numer. XV (1976), 165--168.

*[C] P. A. Catlin, Embedding subgraphs and coloring graphs under extremal degree conditions, Ph. D. Thesis, Ohio State Univ., Columbus (1976).

[KY1] A. V. Kostochka and G. Yu, An Ore-type analogue of the Sauer-Spencer Theorem, Graphs Combin. 23 (2007), 419--424.

[KY2] A. V. Kostochka and G. Yu, An Ore-type graph packing problems, Combin. Probab. Comput. 16 (2007), 167--169.

[SS] N. Sauer and J. Spencer, Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295--302.

* indicates original appearance(s) of problem.