The main result of this paper is that for a special, but rather wide class of "sample graphs", the extremal graphs, i.e, the graphs of n vertices without subgraphs isomorphic to the sample graph and having maximum number of edges under this condition, have very simple and sYmmetric structure. This result remains valid even in the case when the' condition "the graph does not contain the sample g...

Let F ⊂ 2[n] be a family of subsets of {1, 2, . . . , n}. For any poset H, we say F is H-free if F does not contain any subposet isomorphic to H. Katona and others have investigated the behavior of La(n,H), which denotes the maximum size of H-free families F ⊂ 2[n]. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of...

A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stab(H; k) we denote the minimum size among the sizes of all (H; k)-vertex stable graphs. Given an integer q ≥ 2, we prove that, apart of some small values of k, stab(Kq; k) = (2q−3)(k+1). This confirms in the affirmative the conjecture of Dudek et al. [(H, k) stable g...

The thesis consists of two parts. In the first part we study two graph transformations,namely the Kelmans transformation and the generalized tree shift. In the second part of thisthesis we study an extremal graph theoretic problem and its relationship with algebraic graphtheory. The main results of this thesis are the following. • We show that the Kelmans transformation is a very ef...

Given a family of graphs H, the extremal number ex(n,H) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family H as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs Hr such that ex(n,Hr) = Θ(n r). This solves a longstanding problem in the area of extremal graph theory.

The graph bandwidth problem is a well-known NP-complete problem. The relation between size of a graph and bandwidth is very interesting. The minimum size required in G with bandwidth B is denoted as m(n, B) while the graph G of order n and bandwidth B with size m(n, B) is called an extremal graph. This paper provides the minimum size for a graph of odd order n, n ≥ 9, and bandwidth (n + 1)/2, a...

A problem, first considered by Erdős and Gallai [2], was to determine the maximum number of edges in any graph on n vertices if it contains no path with k + 1 vertices. This maximum number, ext(n, Pk+1), is called the extremal number for the path Pk+1. Erdős and Gallai proved the following theorem, which was one of the earliest extremal results in graph theory. Theorem 1.1 ([2]). For every k ≥ ...

Let D = (V,E) be a minimally k-edge-connected simple directed graph. We prove that there is a function f(k) such that |V | ≥ f(k) implies |E| ≤ 2k(|V | − k). We also determine the extremal graphs whose size attains this upper bound.

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a conne...

We determine the number of perfect matchings and forests in a family Tr,3 of triangulated prism graphs. These results show that the extremal number of perfect matchings in a planar graph with n vertices is bounded below by Ω ( 6 √ 7 + √ 37 n) = Ω (1.535) and the extremal number of forests contained in a planar graph with n vertices is bounded below by