Ramsey Number Research Articles

Some results on the multipartite Ramsey numbers mj(C3,Cm,n1K2,n2K2,…,niK2)

The graph Kj×t is a graph which is complete and multipartite which includes j partite sets and t vertices in each partite set. The multipartite Ramsey number (M-R-number) mj(G1,G2,…,Gn) is the smallest integer t for the mentioned graphs G1,G2,…,Gn, in a way which for each n-edge-coloring (G1,G2,…,Gn) of the edges of Kj×t, Gi contains a monochromatic copy of Gi for at least one i. The size of M-R-number mj(nK2,C7) for j≥2, n≤6, the M-R-number mj(nK2,C7) for j=2,3,4, n≥2, the M-R-number mj(nK2,C7) for each j≥5, n≥2, the M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) for j≤6, and i,ni≥1, and the size of M-R-number mj(C3,C3,nK2) for j≥2 and n≥1 have been calculated in various articles hitherto. We acquire some bounds of M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) in this essay in which i,j≥2, and ni≥1, also the size of M-R-number m4(C3,C4,nK2) for each n≥1 is computed in this paper.

Online and Connected Online Ramsey Numbers of a Matching versus a Path

The (G1,G2)-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors it red or blue. The builder aims to force either a red copy of G1 or a blue copy of G2 in as few rounds as possible, while the painter’s aim is the opposite. The online Ramsey number r˜(G1,G2) is the minimum number of edges that the builder needs to win the (G1,G2)-online Ramsey game, regardless of the painter’s strategy. Furthermore, we initiate the study of connected online Ramsey game, which is identical to the usual one, except that at any time the graph induced by all edges should be connected. In this paper, we show a general bound of the online Ramsey number of a matching versus a path and determine its exact value when the path has an order of three or four. For the connected version, we obtain all connected online Ramsey numbers of a matching versus a path.

Computation of new diagonal graph Ramsey numbers

For various connected simple graphs G , we extend the table of diagonal graph Ramsey numbers R ( G , G ) in ‘ An Atlas of Graphs .’ This is accomplished by first converting the calculation of R ( G , G ) into a satisfiability problem in propositional logic. Mathematical arguments and scientific computing are then used to calculate R ( G , G ) .

Multipartite Ramsey Numbers of Union of Stars

Let s and k be positive integers with k ≥ 2 and G 1 , G 2 , …, G k be simple graphs. The set multipartite Ramsey number , denoted by M s ( G 1 , G 2 , …, G k ) , is the smallest positive integer c such that any k -coloring of the edges of K c × s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, …, k } . The size multipartite Ramsey number , denoted by m c ( G 1 , G 2 , …, G k ) , is the smallest positive integer s such that any k -coloring of the edges of K c × s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, …, k } . In this paper, we establish some lower and upper bounds, and some exact values of multipartite Ramsey numbers for the union of stars.

Some Applications of Graph Theory in Middle School Mathematics Teaching

In middle school mathematics, some teaching contents embody a teaching idea of "combination of numbers and graphs", especially in the geometry part of middle school mathematics. Therefore, in the actual process of teaching, teachers should combine numbers and graphs to guide students to learn and explore some practical mathematical problems with intuitive and vivid graphics. Graph theory is a branch of discrete mathematics, which takes graph as its research object. A graph in graph theory is a graph composed of some vertices and edges connecting two vertices. This graph is usually used to describe a specific relationship between certain things. In a graph, a vertex represents a thing, and edges are used to represent specific relationships between connected two vertices. In the history of graph theory, the emergence of graph theory has solved many mathematical problems, especially some seemingly simple problems that are difficult to solve with traditional mathematical methods. The method of graph theory is widely used in middle school mathematics, especially in middle school mathematics competitions. This paper discusses some applications of graph theory in middle school mathematics. We first introduces graph theory through two specific problems in life, and then explores some applications of Ramsey number problem, bipartite graph and matching problem of graph theory in middle school mathematics teaching.

The (t−1) $(t-1)$‐chromatic Ramsey number for paths

AbstractThe following relaxation of the classical problem of determining the Ramsey number of a fixed graph has first been proposed by Erdős, Hajnal and Rado over 50 years ago. Given a graph and an integer determine the minimum number such that in any ‐coloured complete graph on vertices there is a copy of using only edges of some colours. We determine the answer precisely when is a path.

Multicolor bipartite Ramsey numbers for quadrilaterals and stars

For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.

Turán and Ramsey numbers in linear triple systems II

In this paper we continue our studies of Turán and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs in which any two triples intersect in at most one vertex. In [7] the two main problems left open were the Turán number of the wicket and the Ramsey property of the sail. In this paper we present some progress towards both of these problems.

Threshold Ramsey multiplicity for paths and even cycles

The Ramsey number r(H) of a graph H is the minimum integer n such that any two-coloring of the edges of the complete graph Kn contains a monochromatic copy of H. While this definition only asks for a single monochromatic copy of H, it is often the case that every two-edge-coloring of the complete graph on r(H) vertices contains many monochromatic copies of H. The minimum number of such copies over all two-colorings of Kr(H) will be referred to as the threshold Ramsey multiplicity of H. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant c such that the threshold Ramsey multiplicity of a path or an even cycle on k vertices is at least (ck)k. This bound is tight up to the constant c. We prove a similar result for odd cycles in a companion paper.

Star-Critical Gallai–Ramsey Numbers of Graphs

The Gallai–Ramsey number \(gr_{k}(K_{3}: H_{1}, H_{2}, \cdots , H_{k})\) is the smallest integer n such that every k-edge-colored \(K_{n}\) contains either a rainbow \(K_3\) or a monochromatic \(H_{i}\) in color i for some \(i\in [k]\). We define the star-critical Gallai–Ramsey number \(gr_{k}^{*}(K_3: H_{1}, H_{2}, \cdots , H_{k})\) as the smallest integer s such that every k-edge-colored \(K_{n}-K_{1, n-1-s}\) contains either a rainbow \(K_3\) or a monochromatic \(H_{i}\) in color i for some \(i\in [k]\). When \(H=H_{1}=\cdots =H_{k}\), we simply denote \(gr_{k}^{*}(K_{3}: H_{1}, H_{2}, \cdots , H_{k})\) by \(gr_{k}^{*}(K_{3}: H)\). We determine the star-critical Gallai–Ramsey numbers for complete graphs and some small graphs. Furthermore, we show that \(gr_{k}^{*}(K_3: H)\) is exponential in k if H is not bipartite, linear in k if H is bipartite but not a star and constant (not depending on k) if H is a star.

Tighter Bounds on Directed Ramsey Number R(7)

Tournaments are orientations of the complete graph. The directed Ramsey number R(k) is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size k, which we denote by $$ TT _k$$ . We include a computer-assisted proof of a conjecture by Sanchez-Flores in Graphs Combinatorics 14(2), 181–200 (1998), that all $$ TT _6$$ -free tournaments on 24 and 25 vertices are subtournaments of $$ ST _{27}$$ , the unique largest $$ TT _6$$ -free tournament. We also classify all $$ TT _6$$ -free tournaments on 23 vertices. We use these results, combined with assistance from a SAT solver, to obtain the following improved bounds on R(7): $$34 \le R(7) \le 47$$ .

Ramsey Numbers of Trails

We initiate the study of Ramsey numbers of trails. Let $k \geq 2$ be a positive integer. The Ramsey number of trails with $k$ vertices is defined as the the smallest number $n$ such that for every graph $H$ with $n$ vertices, $H$ or the complete $\overline{H}$ contains a trail with $k$ vertices. We prove that the Ramsey number of trails with $k$ vertices is at most $k$ and at least $2\sqrt{k}+\Theta(1)$. This improves the trivial upper bound of $\lfloor 3k/2\rfloor -1$.

Gallai–Ramsey number of odd cycles with chords

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai k-coloring is a Gallai coloring that uses at most k colors. For an integer k≥1, the Gallai–Ramsey number GRk(H) of a given graph H is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of H. Let Cm denote the cycle on m≥4 vertices and let Θm denote the family of graphs obtained from Cm by adding an additional edge joining two non-consecutive vertices. We prove that GRk(Θ2n+1)=n⋅2k+1 for all k≥1 and n≥3. This implies that GRk(C2n+1)=n⋅2k+1 all k≥1 and n≥3. Our result yields a unified proof for the Gallai–Ramsey number of all odd cycles on at least five vertices.

The Ramsey Numbers of Trees Versus Generalized 6-Wheels or Generalized 7-Wheels

The Ramsey Numbers of Trees Versus Generalized 6-Wheels or Generalized 7-Wheels

On Multicolor Ramsey Numbers and Subset Coloring of Hypergraphs

On Multicolor Ramsey Numbers and Subset Coloring of Hypergraphs

Star-Critical Ramsey Numbers Involving Large Generalized Fans

Star-Critical Ramsey Numbers Involving Large Generalized Fans

Gallai—Ramsey Number for the Union of Stars

Given a graph G and a positive integer k, define the Gallai—Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain exact values of the Gallai—Ramsey numbers for the union of two stars in many cases and bounds in other cases. This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.

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 numbers of trails and circuits

AbstractWe show that every two‐colouring of the edges of the complete graph contains a monochromatic trail or circuit of length at least , which is asymptotically best possible.

On size multipartite Ramsey numbers of large paths versus wheel on five vertices

For given two graphs [Formula: see text] and [Formula: see text], and integer [Formula: see text], the size multipartite Ramsey number [Formula: see text] is the smallest integer such that every factorization of graph [Formula: see text] satisfies the following condition: either [Formula: see text] contains [Formula: see text] as a subgraph or [Formula: see text] contains [Formula: see text] as a subgraph. In this paper, we determine that [Formula: see text] for [Formula: see text] where [Formula: see text] is path on [Formula: see text] vertices and [Formula: see text] is a wheel on five vertices.