Cube-Supersaturated Graphs and Related Problems

In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges . Given a graph L, ex(n , L) will denote the maximum number of edges a graph G" of order n can have without containing any L . Determining ex(n,L), or at least finding good bounds on it will be called TURÁN TYPE EXTREMAL PROBLEM. Assume that a graph G" has E > ex(n , L) edges. Then it must contain some copies of L . Such a graph will be called supersaturated, or L-supersaturated . Lsupersaturated graphs mostly contain not only one, but very many copies of L . The problem discussed here (and called "the problem of L-supersaturated graphs") is Determine the minimum number of copies of L a graph G" with E>ex(n,L) edges must contain . The main results of this paper are two "recursion theorems" motivated by the case when L is the graph determined by the vertices and edges of a cube . Notation Below we shall consider ordinary graphs, that is, graphs without loops and multiple edges . G,H, . . . ,S and G",H", . . . , S" will denote graphs and the upper indices will always denote the number of vertices . Also, we shall use v(G), e(G) and X(G) to indicate the number of vertices, edges, and the chromatic number, respectively . Kp is the complete graph on p vertices, Cp is the cycle of length p, K,, , ,, denotes the complete bipartite graph with p and q vertices in its color-classes . G (A , B) denotes the bipartite subgraph of G induced by A and B, (A n B = 0) .