Graf Ramsey (3K2, C3)-Minimal

On Ramsey (3K2,P3)-minimal graphs

For any given two graphs G and H, we use the notation F→(G,H) to mean that in any red-blue coloring of the edges of F , the following must hold: F contains either a red subgraph G or a blue subgraph H. A graph F is a Ramsey (G,H)minimal graph if F→(G,H) but F∗ 6→(G,H) for any proper subgraph F∗ of F . Let R(G,H) be the class of all Ramsey (G,H)-minimal graphs. In this paper, we derive the prope...

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 ...

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...

On Ramsey Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest F other than a matching, and every graph G containing a cycle, there exists an infinite number of graphs J such that J → (F,G) but if we delete from J any edge e the graph J − e obtained in this way does not have this property. Introduction. All graphs in this note are undirected graphs, without loops and multip...

On Ramsey Minimal Graphs