Ramsey-minimal graphs for multiple copies

Ramsey-minimal Graphs for Multiple Copies

Ramsey Theorems for Multiple Copies of Graphs

If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph Kp are colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G . The following result is proved : Let G and H have k and I points respectively and have point independence nu...

All Ramsey (2K2,C4)−Minimal Graphs

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

Ramsey-minimal graphs for forests

It is shown in this paper that the pair (G, H) is Ramsey infinite when both G and H are forests, with at least one of G or H having a non-star component. In addition, an infinite subfamily of R(PP.) is constructed .

Minimal Ordered Ramsey Graphs

An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H ′) of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all e...