Ramsey Number Of Paths Research Articles

Anti-van der Waerden Numbers on Graphs

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. A k-term arithmetic progression of a graph G (k-AP) is a list of k distinct vertices such that the distance between consecutive pairs is constant. A rainbow k-AP is a k-AP where each vertex is colored distinctly. This allows for the definition of the anti-van der Waerden number of a graph G, which is the least positive integer r such that every exact r-coloring of G contains a rainbow k-AP. Much of the focus of this paper is on 3-term arithmetic progressions for which general bounds are obtained based on the radius and diameter of a graph. The general bounds are improved for trees and Cartesian products and exact values are determined for some classes of graphs. Longer k-term arithmetic progressions are considered and a connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established.Please confirm if the inserted city and country name for all affiliations is correct. Amend if necessary.The cities and affiliations are correct.

Ramsey number of paths and connected matchings in Ore-type host graphs

It is well-known (as a special case of the path–path Ramsey number) that in every 2-coloring of the edges of K3n−1, the complete graph on 3n−1 vertices, there is a monochromatic P2n, a path on 2n vertices. Schelp conjectured that this statement remains true if K3n−1 is replaced by any host graph on 3n−1 vertices with minimum degree at least 3(3n−1)4. Here we propose the following stronger conjecture, allowing host graphs with the corresponding Ore-type condition: If G is a graph on 3n−1 vertices such that for any two non-adjacent vertices u and v, dG(u)+dG(v)≥32(3n−1), then in any 2-coloring of the edges of G there is a monochromatic path on 2n vertices. Our main result proves the conjecture in a weaker form, replacing P2n by a connected matching of size n. Here a monochromatic, say red, matching in a 2-coloring of the edges of a graph is connected if its edges are all in the same connected component of the graph defined by the red edges. Applying the standard technique of converting connected matchings to paths with the Regularity Lemma, we use this result to get an asymptotic version of our conjecture for paths.

Ramsey Numbers of Squares of Paths

The Ramsey number R(G,H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G,H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power Pnr of a path on n vertices is obtained by joining any two vertices with distance at most r.We show that R(P3n2,P3n2)=9n−3 for n large and discuss some related questions.

The Ramsey number of paths with respect to wheels

For graphs G and H , the Ramsey number R ( G , H ) is the smallest positive integer n such that every graph F of order n contains G or the complement of F contains H . For the path P n and the wheel W m , it is proved that R ( P n , W m ) = 2 n - 1 if m is even, m ⩾ 4 , and n ⩾ ( m / 2 ) ( m - 2 ) , and R ( P n , W m ) = 3 n - 2 if m is odd, m ⩾ 5 , and n ⩾ ( m - 1 / 2 ) ( m - 3 ) .

Undecided Ramsey-Numbers for Paths

Undecided Ramsey-Numbers for Paths