Let P be a non-trivial hereditary property of graphs and let k the minimum chromatic number graph that does not belong to P. We prove that, for every fixed p∈(0,1), maximum possible edges in subgraph random G(n,p) which belongs is, with high probability,(1−1k−1+o(1))p(n2).

For a countable, weakly minimal theory T , we show that the SchröderBernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following: 1. For any U -rank-1 type q ∈ S(acl(∅)) and any automorphism f of the monster model C, there is some n < ω such that f(q) is not almost orthogonal to q ⊗ f(q)⊗ . . .⊗ fn−1(q); 2. T has no infinite collection of ...

We prove that a hereditary property of cographs has bounded linear cliquewidth if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permutation classes, and the similarities between these two strands of investigation lead us to a conjecture relating the graph properties of bounded linear clique-width to permutation c...

For a graphical property P and a graph G, a subset S of vertices of G is a P-set if the subgraph induced by S has the property P . The domination number with respect to the property P , is the minimum cardinality of a dominating P-set. In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate and hereditary properties when a graph is ...

For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [19] show that any non-constant monotone property P : {0, 1}( n 2) → {0, 1} of n-vertex graphs has D(P) = Ω(n). We extend their result to 3-uniform hypergraphs. In particular, we ...

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let P and Q be additive hereditary properties of graphs. The generalized chromatic number χQ(P) is defined as follows: χQ(P) = n iff P ⊆ Qn but P 6⊆ Qn−1. We investigate the generalized chromatic numbers of the well-known properties of graphs Ik, Ok, Wk, Sk and Dk.

For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the largest possible edit distance of a graph on n vertices from P? Denote this distance by ed(n,P). A graph property is hereditary if it is closed under removal of vertic...

Given a graph property P, it is interesting to determine the typical structure of graphs that satisfy P. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if P is a monotone property and r is the largest integer for which every r-colorable graph satis es P, then almost every gra...