What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Topological minors in bipartite graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K _(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.     

The number of edges of radius-invariant graphs

The eccentricity e(υ) of vertex υ is defined as a distance to a farthest vertex from υ. The radius of a graph G is defined as r(G) = {e(u)}. We consider properties of unchanging the radius of a graph under two different situations: deleting an arbitrary edge and deleting an arbitrary vertex. This paper gives the upper bounds for the number of edges in such graphs. Supported by VEGA grant No. 1/0084/08.     

Almost Given Length Cycles in Digraphs

A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every ε > 0 there are constants K and δ so that for every d ∈ (0, δn), every ε n²-cyclic digraph with n vertices contains a directed cycle whose length is between d and d + K. A more general result of the same form is obtained for blow-ups of directed cycles.