On path-sunflower Ramsey numbers (original) (raw)
On some Ramsey and Turán-type numbers for paths and cycles
For given graphs G 1 , G 2 , ..., G k , where k ≥ 2, the multicolor Ramsey number R(G 1 , G 2 , ..., G k ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, there is always a monochromatic copy of G i colored with i, for some 1 ≤ i ≤ k. Let P k (resp. C k ) be the path (resp. cycle) on k vertices. In the paper we show that R(P 3 , C k , C k ) = R(C k , C k ) = 2k − 1 for odd k. In addition, we provide the exact values for Ramsey numbers R(P 4 , P 4 , C k ) = k + 2 and R(P 3 , P 5 , C k ) = k + 1.
On some three-color Ramsey numbers for paths
Discrete Applied Mathematics, 2015
For graphs G 1 , G 2 , G 3 , the three-color Ramsey number R(G 1 , G 2 , G 3) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ 3. First, we prove that the conjectured equality R(C 2n , C 2n , C 2n) = 4n, if true, implies that R(P 2n+1 , P 2n+1 , P 2n+1) = 4n + 1 for all n ≥ 3. We also obtain two new exact values R(P 8 , P 8 , P 8) = 14 and R(P 9 , P 9 , P 9) = 17, furthermore we do so without help of computer algorithms. Our results agree with a formula R(P n , P n , P n) = 2n−2+(n mod 2) which was proved for sufficiently large n by Gyárfás, Ruszinkó, Sárközy, and Szemerédi in 2007. This provides more evidence for the conjecture that the latter holds for all n ≥ 1.
Tripartite Ramsey numbers for paths
Journal of Graph Theory, 2007
In this article, we study the tripartite Ramsey numbers of paths. We show that in any two‐coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 − o(1))2n. Since R(P2n+1,P2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two‐colorings of K3n and K(n, n, n) are considered. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 164–174, 2007
Combinatorics, Probability and Computing, 2009
For two graphs S and T , the constrained Ramsey number f (S, T ) is the minimum n such that every edge coloring of the complete graph on n vertices (with any number of colors) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T . Here, a subgraph is said to be rainbow if all of its edges have different colors. It is an immediate consequence of the Erdős-Rado Canonical Ramsey Theorem that f (S, T ) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f (S, T ) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f (S, T ) ≤ O(st 2 ) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f (S, P t ) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.
On the Ramsey numbers for paths and generalized Jahangir graphs
Bulletin Mathematiques De La Societe Des Sciences Mathematiques De Roumanie, 2008
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(tP n , H), where H is a generalized Jahangir graph J s,m where s ≥ 2 is even, m ≥ 3 and t ≥ 1 is any integer.
Graphs and Combinatorics, 2015
For two given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G contains a G 2. In this note, we determined the Ramsey number R(K 1,n , W m) for even m with n + 2 ≤ m ≤ 2n − 2, where W m is the wheel on m + 1 vertices, i.e., the graph obtained from a cycle C m by adding a vertex v adjacent to all vertices of the C m .
On the Union of Graphs Ramsey Numbers
Let H be a graph with the chromatic number χ(H) and the chromatic surplus σ(H). A connected graph G of order n is called good with respect to H, denoted by H-good, if R(G, H) = (n−1)(χ(H)−1)+σ(H). In this paper, we investigate the Ramsey numbers for a union of graphs not necessarily containing an H-good component.
The Ramsey number of paths with respect to wheels
Discrete Mathematics, 2005
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) = 2n − 1 if m is even, m 4, and n (m/2)(m − 2), and R(P n , W m) = 3n − 2 if m is odd, m 5, and n (m − 1/2)(m − 3).
On size multipartite Ramsey numbers for paths versus stars
International Journal of Mathematical Analysis, 2016
For given two graphs G 1 and G 2 , and integer j ≥ 2, the size multipartite Ramsey numbers m j (G 1 , G 2) = t is the smallest integer such that every factorization of graph K j×t := F 1 ⊕ F 2 satisfies the following condition: either F 1 contains G 1 as a subgraph or F 2 contains G 2 as a subgraph. In this paper, we determine that m j (K 1,m , P l) for l ≥ 2 where K 1,n is a star on n vertices and P l is a path on l vertices.
Ramsey numbers for graph sets versus complete graphs
1996
Agsrnacr. The set Ramsey number r.(t, (l)) is the smallest integer r such that if the edges of a complete graph K, are 2-colored, then there will be a graph with n vertices a,nd /c edges in the first cblorbr a graph with n vertices ana (|) (e.g. a complete graph) in the second eolor. For each n ) 3 and 1 < k < n, the set Ramsey numbers r.(f, (l)) are determined. One approach to get some insight into r(K") was suggested in , where Ramsey numbers for sets of graphs with fixed numbers of vertices and edges were considered. Thus, the following definition. Definition 1. For positive integers n'L,r\s,t with 1 ( s ( (\) ana t < t < (Z) the set Ramsey number r^,n(s,t) is the smallest integer r such that for arry 2-coloring of the edges of a complete graph K,, there is either a graph with m vertices and's edges in the first color, or a graph with n vertices and t edges in the second color. When rrl : rL, r*,^(s,t) will be expressed more compactly as r*(s,t). Associated with fixed positive integers rn and n there is an (!) UV (i) array of Ramsey numbers (r*,n(t,t)) for 1 ( s ( (f) ana 1 <, < (!) ttrat represent sets of graphs with fixed numbers of edges. For small values of m and n the array of Ramsey numbers (r*,n(t,t)) have been determined. The values for m :3 and 3 I n 17 were determined in [4], and the values for m: 4 and 4 1n 15 were determined in , except ior ra,s((l), (!)) : r(Ka, K5), which has now been shown to be 25 by McKay and R^adziszowski (see [fl). All but 5 cf the values of (r5(s,t)) were determined in [5]. For Ramsey numbers of more general sets of small graphs see [2].