Abstract The size-Ramsey number of a graph F is the smallest edges in G with Ramsey property for , that is, any 2-colouring contains monochromatic copy . We prove grid on n × vertices bounded from above by 3+o(1)

Given a graph G and positive integer k, define the Gallai—Ramsey number to be minimum of vertices n such that any k-edge coloring Kn contains either rainbow (all different colored) triangle or monochromatic copy G. In this paper, we obtain exact values numbers for union two stars in many cases bounds other cases. This work represents first class disconnected graphs considered as desired subgraph.

Given a labeled graph H with vertex set {1, 2, . . . , n}, the ordered Ramsey number r<(H) is the minimum N such that every two-coloring of the edges of the complete graph on {1, 2, . . . , N} contains a copy of H with vertices appearing in the same order as in H. The ordered Ramsey number of a labeled graph H is at least the Ramsey number r(H) and the two coincide for complete graphs. However,...

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 improve the Surahmat and Tomescu’s result [9] on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number R(∪G, H), where G is a path and H is a Jah...

For a given graph $H$, the Ramsey number $r(H)$ is minimum $N$ such that any 2-edge-coloring of complete $K_N$ yields monochromatic copy $H$. Given positive integer $n$, \emph{fan }$F_n$ formed by $n$ triangles share one common vertex. We show ${9n}/{2}-5\le r(F_n)\le {11n}/{2} + 6$ for $n$. This improves previous best bounds $r(F_n) \le 6n$ Lin and Li \ge 4n+2$ Zhang, Broersma Chen.

A Gallai coloring is a of the edges complete graph without rainbow triangles, and k-coloring that uses at most k colors. For an integer k≥1, Gallai–Ramsey number GRk(H) given H least positive N such every KN contains monochromatic copy H. Let Cm denote cycle on m≥4 vertices let Θm family graphs obtained from by adding additional edge joining two non-consecutive vertices. We prove GRk(Θ2n+1)=n⋅2...