Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Some Known Results and an Open Problem of Tree - Wheel Graph Ramsey Numbers

There are many famous problems on finding a regular substructure in a sufficiently large combinatorial structure, one of them i.e. Ramsey numbers. In this paper we list some known results and an open problem on graph Ramsey numbers. In the special cases, we list to determine graph Ramsey numbers for trees versus wheel

Some Known Results and an Open Problem of Tree - Wheel Graph Ramsey Numbers

There are many famous problems on finding a regular substructure in a sufficiently large combinatorial structure, one of them i.e. Ramsey numbers. In this paper we list some known results and an open problem on graph Ramsey numbers. In the special cases, we list to determine graph Ramsey numbers for trees versus wheel

Some exact values on Ramsey numbers related to fans

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an positive integer $n$, let $K_{1,n}$ be a star with $n+1$ vertices, $F_n$ be a fan with $2n+1$ vertices consisting of $n$ triangles sharing one common vertex, and $nK_3$ be a graph with $3n$ vertices obtained from the disjoint union of $n$ triangles. In 1975, Burr, Erd\H{o}s and Spencer \cite{B} proved that $R_2(nK_3)=5n$ for $n\ge2$. However, determining the exact value of $R_2(F_n)$ is notoriously difficult. So far, only $R_2(F_2)=9$ has been proved. Notice that both $F_n$ and $nK_3$ contain $n$ triangles and $|V(F_n)|<|V(nK_3)|$ for all $n\ge 2$. Chen, Yu and Zhao (2021) speculated that $R_2(F_n)\le R_2(nK_3)=5n$ for $n$ sufficiently large. In this paper, we first prove that $R(K_{1,n},F_n)=3n-\varepsilon$ for $n\ge1$, where $\varepsilon=0$ if $n$ is odd and $\varepsilon=1$ if $n$ is even. Applying the exact values of $R(K_{1,n},F_n)$, we will confirm $R_2(F_n)\le 5n$ for $n=3$ by showing that $R_2(F_3)=14$.Comment: 10 pages, 3 figure

On the Tur\'{a}n number of Km∨C2k−1K_m \vee C_{2k-1}

Given a graph $H$ and a positive integer $n$, the Tur\'{a}n number of $H$ for the order $n$, denoted $ex(n,H)$, is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. Given graphs $G$ and $H$, the notation $G \vee H$ means the joint of $G$ and $H$. $\chi(G)$ denotes the chromatic number of a graph $G$. Since $\chi(K_m \vee C_{2k-1})=m+3$ and there is an edge $e\in E(K_m \vee C_{2k-1})$ such that $\chi(K_m \vee C_{2k-1}-e)= m+2$, by the Simonovits theorem, $ex(n, K_m \vee C_{2k-1}) = \lfloor \frac{(m+1)n^2}{2(m+2)}\rfloor$ for sufficiently large $n$. In this paper, we prove that $2(m+2)k-3(m+2)-1$ is large enough for $n$.Comment: 5 page

Ramsey numbers involving a triangle: theory and algorithms

Ramsey theory studies the existence of highly regular patterns in large sets of objects. Given two graphs G and H, the Ramsey number R(G, H) is defined to be the smallest integer n such that any graph F with n or more vertices must contain G, or F must contain H. Albeit beautiful, the problem of determining Ramsey numbers is considered to be very difficult. We focus our attention on efficient algorithms for determining Ram sey numbers involving a triangle: R(K3 , G). With the help of theoretical tools, the search space is reduced by using different pruning techniques and linear programming. Efficient operations are also carried out to mathematically glue together small graphs to construct larger critical graphs. Using the algorithms developed in this thesis, we compute all the Ramsey numbers R(Kz,G), where G is any connected graph of order seven. Most of the corresponding critical graphs are also constructed. We believe that the algorithms developed here will have wider applications to other Ramsey-type problems

The Ramsey numbers for Disjoint Union of trees versus W4

The Ramsey number for a graph G versus a graph H, de-\ud noted by R(G;H), is the smallest positive integer n such that for any\ud graph F of order n, either F contains G as a subgraph or F contains H as\ud a subgraph. In this paper, we investigate the Ramsey numbers for union\ud of trees versus small cycle and small wheel. We show that if ni is odd and\ud 2ni+1 ?? ni for every i, then R(\ud Sk\ud i=1 Tni ;W4) = R(Tnk ;W4) +\ud Pk??1\ud i=1 ni\ud for k ?? 1:. Furthermore, we show that\ud 1. If ni is even and 2ni+1 ?? ni + 1 for every i, then R(\ud Sk\ud i=1 Sni ;W4) =\ud 2nk +\ud Pk??1\ud i=1 ni for k ?? 2;\ud 2. If ni is odd and 2ni+1 ?? ni for every i, then R(\ud Sk\ud i=1 Sni ;W4) =\ud R(Snk ;W4) +\ud Pk??1\ud i=1 ni for k ?? 1

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement