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...

For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is least $N$ such that every $k$-coloring of edges complete $K_N$ contains monochromatic copy $H$. Let $C_m$ denote cycle on $m\ge4$ vertices let $\Theta_m$ family graphs obtained from by adding additional edge joining two non-consecutive vertices. Unlike odd cycles, little known about general behavior $R_k(C_{2n})$ ex...

A = {(1, 0), (2, 0), . . . , (n, 0)}, B = {((1, 1), (2, 1), . . . , (n, 1)} and the edge ab is the line segment joining a ∈ A and b ∈ B in R. This model is essentially the same as the cyclic bipartite graphs and ordered bipartite graphs considered earlier by several authors. Subgraphs — paths, trees, double stars, matchings — are called non-crossing if they do not contain edges with common inte...

The Ramsey number r(H, Kn) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turán number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H . We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to ...