Extremal problems for directed graphs

Extremal Problems for Directed Graphs

We consider directed graphs without loops and multiple edges, where the exclusion of multiple edges means that two vertices cannot be joined by two edges of the same orientation. Let L I ,. . ., LQ be given digraphs. What is the maximum number of edges a digraph can have if it does not contain any L i as a subgraph and has given number of vertices? We shall prove the existence of a sequence of ...

Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs

Notation. Given a graph, hypergraph Gn, . . . , the upper index always denotes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G respectively. Given a family L of graphs, hypergraphs, ex(n,L) denotes the maximum number of edges (hyperedges) a graph (hypergraph)Gn of order n can have without containing subgraphs (subhypergraphs) from L...

Extremal problems concerning Kneser graphs

Let d and S? be two intersecting families of k-subsets of an n-element set. It is proven that l~.JuS?l <(;:i)+(;::) holds for n>f(3+,/?)k, and equality holds only if there exist two points a, b such that {a, b} n F# 0 for all FE d u g, For n=2k+o(Jj;) an example showing that in this case max 1 d u B 1 = (1 o( 1 ))( ;) is given. This disproves an old conjecture of Erdiis [7]. In the second part ...

Extremal problems for local properties of graphs

Some Extremal Problems for Edge-Regular Graphs

We consider the class ER(n, d, λ) of edge-regular graphs for some n > d > λ, i.e., graphs regular of degree d on n vertices, with each pair of adjacent vertices having λ common neighbors. It has previously been shown that for such graphs with λ > 0 we have n ≥ 3(d − λ) and much has been done to characterize such graphs when equality holds. Here we show that n ≥ 3(d − λ) + 1 if λ > 0 and d is od...