Spanning Trees in Dense Graphs
Combinatorics, probability & computing
In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n 0 such that, if n > n 0 , T is a tree of order n and maximum degree at most cn/ log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G. Given a rooted tree and a vertex v, we write A(v) for the set of ancestors of v, C(v) for the set of children of v, G(v) for the set of grandchildren of v, and T (v) for the set of descendants of v (including v
... of v (including v itself). In a tree T we denote the set of leaves by L(T ). In a star S we denote the middle vertex (or the root) by M(S). A rooted forest is a forest of rooted trees. We will use a b to denote that a is sufficiently small compared to b. For simplicity, we do not always compute these dependences, although it could be done. Packings and subtrees in dense graphs Definition 3. An embedding of a graph G = (V , E) into a graph G = (V , E ) is an edge-preserving one-to-one map from V to V , that is, an injection ϕ : Such a map ϕ induces an injection from E into E ; we will use the notation ϕ for that map, too. In particular, we will write ϕ(E) for the image set of the edges of G. Definition 4. Given a set of graphs G 1 , G 2 , . . . , G l , we say that G 1 , G 2 , . . . , G l can be packed into G if we can find embeddings ϕ i of G i into G such that the edge sets ϕ i (E(G i )) are pairwise disjoint. If G = K n , the complete graph on n vertices, then we say simply that there is a packing of G 1 , G 2 , . . . , G l . The notion of packing plays an important role in the investigation of computational complexity of graph properties. Thus it is not surprising that in recent research literature there is considerable interest in packing-type results and problems (see, e.g., [2, 3, 4, 13] ). Along these lines, solving an old conjecture of Bollobás  we proved the following . Theorem 1.1. Let ∆ and c < 1/2 be given. Then there exists a constant n 0 with the following properties. If n > n 0 , T is a tree of order n with ∆(T ) 6 ∆, and G is a graph of order n with ∆(G) 6 cn, then there is a packing of T and G. We proved this theorem in the following equivalent embedding form. Theorem 1.1 . Let ∆ and δ > 0 be given. Then there exists a constant n 0 with the following properties. If n > n 0 , T is a tree of order n with ∆(T ) 6 ∆, and G is a graph of order n with δ(G) > ((1/2) + δ)n, then T is a subgraph of G.