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 ...

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 ...

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...

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...

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these 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...

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.

A graph G is r-ramsey-minimal with respect to Kk if every rcolouring of the edges of G yields a monochromatic copy of Kk, but the same is not true for any proper subgraph of G. In this paper we show that for any integer k ≥ 3 and r ≥ 2, there exists a constant c > 1 such that for large enough n, there exist at least c 2 non-isomorphic graphs on at most n vertices, each of which is r-ramsey-mini...

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...

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H ′ are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H ′. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph Kk. A famous theorem of Nešetřil and Rödl implies that any graph H which ...