For two given graphs G1 and G2, the Ramsey number r(G1, G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Km denote a complete graph of order m and Kn −P3 a complete graph of order n without two incident edges. In this paper, we prove that r(K5 − P3,K5) = 25 without help of computer algorithms.

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...

The lower bound for the classical Ramsey number R(4, 6) is improved from 35 to 36. The author has found 37 new edge colorings of K35 that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found ...

For given graphs G and H, the Ramsey number R(G, H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number R(tPn, H), where H is a generalized Jahangir graph Js,m where s ≥ ...