On Ramsey Minimal Graphs
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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 ...
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For graphs F and H, we say F is Ramsey for H if every 2-coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H-minimal if there is no proper subgraph F ′ of F so that F ′ is Ramsey for H. Burr, Erdős, and Lovász defined s(H) to be the minimum degree of F over all Ramsey H-minimal graphs F . Define Ht,d to be a graph on t+ 1 vertices consisting of a complete graph...
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For graphs F , G and H , we write F → (G, H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H . The graph F is Ramsey (G, H)-minimal if F → (G, H) but F ∗ 9 (G, H) for any proper subgraph F ∗ ⊂ F . We present an infinite family of Ramsey (K1,2, C4)-minimal graphs of any diameter ≥ 4.
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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...
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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...
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For graphs F and H, we say F is Ramsey for H if every 2-coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H-minimal if F is Ramsey for H and there is no proper subgraph F ′ of F so that F ′ is Ramsey for H. Burr, Erdős, and Lovász defined s(H) to be the minimum degree of F over all Ramsey H-minimal graphs F . Define Ht,d to be a graph on t + 1 vertices consist...
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