Packing two bipartite graphs into a complete bipartite graph

Abstract

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this paper, we prove that any two compatible C4-free bipartite graphs of order n together with at most 2n - 2 edges can be packed into a complete bipartite graph of order at most n + 1, unless one is the union of vertex-disjoint cycles and the other is the union of two vertex-disjoint stars. This theorem fails for non-C4-free bipartite graphs. We will provide a family of non-C4-free bipartite graphs to serve as examples. As in Wang and Sauer (1993) (H. Wang and N. Sauer, Packing three copies of a tree into a complete graph, Eur. J. Combinatorics 14 (1993), 137–142) for each of infinitely many n, there is a pair of compatible forests of order n which cannot be packed into a complete bipartite graph of order n. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 95–104, 1997