Packing two copies of a sparse graph into a graph with restrained maximum degree

Abstract

Abstract:We show that if a tree T is not a star, then there is an embedding of T in the complement of T such that the maximum degree of T∪ (T) is at most (T)+2. We also show that if G is a graph of order n with n−1 edges, then with several exceptions, there exists an embedding of G in the complement of G such that the maximum degree of G∪ (G) is at most (G)+3. Both results are sharp in the sense that neither of (T)+2 and (G)+3 can be reduced. From these two results, we deduce two corollaries on packings of three graphs. ᭧