All graphs in this paper are understood to be finite, undirected, without loops or multiple edges. The graph G' = (V', E') is called an induced subgraph of G = (V, E) if V' ~ V and uv E E' if and only if {u, v} ~ V', uv E E. The following two problems about induced matchings have been formulated by Erdos and Nesetril at a seminar in Prague at the end of 1985: 1. Determine f(k, d), the maximum n...

We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n log n upper bound on the ordered graph extremal function of F = ({1, 3}, {1, 5}, {2, 3}, {2, 4}) due to Füredi to the n(log n)2(log log n)3 upper bound in the hypergraph case. We use Davenp...

A dominating set S of a graph G is a global (strong) defensive alliance if for every vertex v ∈ S, the number of neighbors v has in S plus one is at least (greater than) the number of neighbors it has in V \ S. The dominating set S is a global (strong) offensive alliance if for every vertex v ∈ V \ S, the number of neighbors v has in S is at least (greater than) the number of neighbors it has i...

Let us write f(n, ∆; C2k+1) for the maximal number of edges in a graph of order n and maximum degree ∆ that contains no cycles of length 2k + 1. For n 2 ≤ ∆ ≤ n − k − 1 and n sufficiently large we show that f(n, ∆; C2k+1) = ∆(n −∆), with the unique extremal graph a complete

The generalization of classical results about convex sets in Rn to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P3-convexity on graphs. A set R of vertices of a graph G is P3-convex if no vertex in V (G) \ R has two neighbours in R. The P3-convex hull of a set of vertices is t...

My research uses algebraic and geometric methods to prove theorems in extremal combinatorics. Going the other way, I also use combinatorial methods to prove algebraic results. Algebraic methods are deeply embedded in my work and nearly all of my success in graph theoretic research has come from attacking purely combinatorial problems through the lens of algebra, combinatorial number theory, or ...

Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G\ the number of vertices. A well-known conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 ~*. This latter number is the proportion of monochromatic Kt's in a random colouring of Kn. We present counterexamples to this conjecture and discuss properties ...

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in Kr,s-interval minor free bipartite graphs. We determine exact values when r = 2 and describe the extremal graphs. For r = 3, lower and upper bounds are given and the structure of K3,s-interval minor free graphs is studied.

In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the k-decomposition for the reciprocal sum-degree distance. Finally,we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs.

Let I(G) be a topological index of a graph. If I(G+ e) < I(G) (or I(G+ e) > I(G), respectively) for each edge e ∈ G, then I(G) decreases (or increases, respectively) with addition of edges. In this paper, we determine the extremal values of some topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs in bipartite graphs with a gi...