On generalized graphs

Abstract

A generalizect graph consists of a set of n vertices and a collection of k-tuples. of these vertices (cf. TURAN [1]). In what follows we shall refer to such a configuration as an edge-grapk if k = 2 and, usually, simply as a graph if k > 2. A complete m-graph has mvertices and [k)k-tuples. We say that a graph G is m-saturated if it contains no complete m-graph but loses this property when any new k-tuple is added.Tu~AN [2] proved the following theorem on edge-graphs in 1941: Let n = = g(m-1)+r, where g, m, and r are integers such that g=>l, m=>3, O<=r<=m -1, and n ~m. Then an m-satfirated edge-graph of n vertices can have at most Em= 2(in 1) (ha -r2) § edges. The dual problem was recently solved by ERI~6S, HAJNAL, and MOON [3] who showed that such an edge-graph must have at least e,,,=(m-2)(n-m § 1) edges. These two results can be combined as follows: If G is an m-saturated edge--graph of n Vertices and e edges, then em ~ e -<_ E,,,.The extremal edge-graphs for which e = % or e = E,~, are also characterized in these papers.Corresponding problems can be stated for generalized graphs. Let G be a (k +/)-saturated graph with n vertices and t k-tuples, where it is understood that k + l~ n. Let the maximum and minimum values of t over the class of such graphs G be denoted by Tk, z and t~.t, respectively.

The saturation function of complete partite graphs

“…Bollobás [2] completely solved the problem for W k 1,...,1 ∼ = S k 1,...,1 . The saturation function sat(n, W 3 1,1,m ) was asymptotically determined in [22] with the result being exact for infinitely many values of n for every m (when Steiner triple systems with certain parameters exist).…”

“…Construction 2 Let r ≥ 2, 1 ≤ s 1 ≤ · · · ≤ s r−1 = s r , s r ≥ 2, and F = K s 1 ,...,sr . Define p by (2).…”

“…Let F = K s 1 ,...,sr and define p by (2). Let n → ∞ and let G be an arbitrary F -saturated graph of order n and size sat(n, F ) + O(1).…”

Transference for the Erdős–ko–rado Theorem

“…We will also need the following well-known inequality for cross-intersecting families due to the second author [6]. Finally, we shall require a theorem of Kruskal [21] and Katona [16].…”

Treewidth Computation and Extremal Combinatorics

“…Then the combinatorial lemma states that t(b, f ) ≤ b+f b (and it is easy to check that this bound is tight). This can be seen as a variation of Bollobáss Theorem [7], which is one of the corner-stones in extremal set theory. (See Section 9.2.2 of [21] for detailed discussions on Bollobáss Theorem and its variants.…”