On the Ramsey number r(H + Kn, Kn)

The drawing Ramsey number Dr(Kn)

Bounds are determined for the smallest m = Dr(Kn) such that every drawing of Km in the plane (two edges have at most one point in common) contains at least one drawing of Kn with the maximum number (:) of crossings. For n = 5 these bounds are improved to 11 :::; Dr(K5) 113. A drawing D( G) of a graph G is a special realization of G in the plane. The vertices are mapped into different points of ...

On Ramsey (K1, 2, Kn)-minimal graphs

Let F be a graph and let G, H denote nonempty families of graphs. We write F → (G,H) if in any 2-coloring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (G,H)-minimal graph if F → (G,H) and F − e 6→ (G,H) for every e ∈ E(F ). We present a technique ...

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

On the Biplanar Crossing Number of Kn

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the Euclidean plane. The k-planar crossing number crk(G) of G is min{cr(G1) + cr(G2) + . . .+ cr(Gk)}, where the minimum is taken over all possible decompositions of G into k subgraphs G1, G2, . . . , Gk. The problem of computing the crossing number of complete graphs, cr(Kn), exactly for sm...

On the Ramsey number R(Cn or Kn-1, Km) (m=3, 4)

The Ramsey number R(Cn or Kn1 , Km) is the smallest integer p such that every graph G on p vertices contains either a cycle en with length n or a K n 1 , or an independent set of order m. In this paper we prove that R(Cn or K n b K 3 ) = 2(n 2) + 1 (n 2: 5), R(Cn or K n 1 , K4) = 3(n 2) + 1 (n 2: 7). In particular, we prove that R(C4 or K 3 , K 3 ) = 6, R(C4 or K 3 , K4) = 8, R(C5 or K4, K 4 ) ...