On path-sunflower Ramsey numbers (original) (raw)
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arXiv: General Mathematics, 2006
In this paper we define new numbers called the Neo-Ramsay numbers. We show that these numbers are in fact equal to the Ramsay numbers. Neo-Ramsey numbers are easy to compute and for finding them it is not necessary to check all possible graphs but enough to check only special kind of graphs having a well-defined adjacency pattern.
On a variation of the Ramsey number
Transactions of the American Mathematical Society, 1972
Let c ( m , n ) c(m,n) be the least integer p p such that, for any graph G G of order p p , either G G has an m m -cycle or its complement G ¯ \bar G has an n n -cycle. Values of c ( m , n ) c(m,n) are established for m , n ⩽ 6 m,n \leqslant 6 and general formulas are proved for c ( 3 , n ) , c ( 4 , n ) c(3,n),c(4,n) , and c ( 5 , n ) c(5,n) .
On the Ramsey numbers for a combination of paths and Jahangirs
2007
For given graphs GGG and H,H,H, the \emph{Ramsey number} R(G,H)R(G,H)R(G,H) is the least natural number nnn such that for every graph FFF of order nnn the following condition holds: either FFF contains GGG or the complement of FFF contains H.H.H. In this paper, we improve the Surahmat and Tomescu's result \cite{ST:06} on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number R(cupG,H)R(\cup G,H)R(cupG,H), where GGG is a path and HHH is a Jahangir graph.
Journal of Graph Theory, 1977
In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs Fl, F, of graphs where F, has at most four points and F2 has exactly five points. Exact results are listed for almost all of these pairs. Let FI, F2 be graphs without isolated points. As in [3], the Ramsey number r(Fl, F2) is the smallest n such that, for every graph G with n 9 7 7 8 18 9 7
On Ramsey numbers of path versus wheel-like graphs
2008
The study of classical Ramsey numbers R(mn) shows little progress in the last two decades. Only nine classical Ramsey numbers are known. This diculty ofnding the classical Ramsey numbers has inspired many people to study generalizations of classical Ramsey number. One of ...
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
On the Ramsey numbers for paths and generalized Jahangir graphs
arXiv (Cornell University), 2008
For given graphs GGG and H,H,H, the \emph{Ramsey number} R(G,H)R(G,H)R(G,H) is the least natural number nnn such that for every graph FFF of order nnn the following condition holds: either FFF contains GGG or the complement of FFF contains H.H.H. In this paper, we determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number R(tPn,H)R(tP_n,H)R(tPn,H), where HHH is a generalized Jahangir graph Js,mJ_{s,m}Js,m where sgeq2s\geq2sgeq2 is even, mgeq3m\geq3mgeq3 and tgeq1t\geq1tgeq1 is any integer.
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.