منابع مشابه
The vertex size-Ramsey number
In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted by R̂v(G, r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ωr(∆n) = R̂v(G, r) = Or(n ) for any G of order n and maximum deg...
متن کاملThe Size-ramsey Number
The size-Ramsey number of a graph G is the smallest number of edges in a graph Γ with the Ramsey property for G, that is, with the property that any colouring of the edges of Γ with two colours (say) contains a monochromatic copy of G. The study of size-Ramsey numbers was proposed by Erdős, Faudree, Rousseau, and Schelp in 1978, when they investigated the size-Ramsey number of certain classes o...
متن کاملThe Size Ramsey Number
Let i2 denote the class of all graphs G which satisfy G-(Gl, GE). As a way of measuring r inimality for members of P, we define the Size Ramsey number ; We then investigate various questions concerned with the asymptotic behaviour of r .
متن کاملThe size-Ramsey number of trees
Given a graph G, the size-Ramsey number r̂(G) is the minimum number m for which there exists a graph F on m edges such that any two-coloring of the edges of F admits a monochromatic copy of G. In 1983, J. Beck introduced an invariant β(·) for trees and showed that r̂(T ) = Ω(β(T )). Moreover he conjectured that r̂(T ) = Θ(β(T )). We settle this conjecture by providing a family of graphs and an emb...
متن کاملTwo variants of the size Ramsey number
Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = maxF⊆G |E(F )|/|V (F )| and define the Ramsey density minf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper we show that when H is a complete graph ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2016
ISSN: 0012-365X
DOI: 10.1016/j.disc.2016.02.001