Defective Ramsey Numbers in Graph Classes

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.

2021

A set of vertices X ⊆ V in a simple graph G(V, E) is irredundant if each vertex x ∈ X is either isolated in the induced subgraph X or else has a private neighbor y ∈ V \ X that is adjacent to x and to no other vertex of X. The irredundant Ramsey number s(m, n) is the smallest N such that in every red-blue coloring of the edges of the complete graph of order N , either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element irredundant set. The mixed Ramsey number t(m, n) is the smallest N for which every red-blue coloring of the edges of K N yields an m-element irredundant set in the blue subgraph or an n-element independent set in the red subgraph. In this paper, we first improve the upper bound of t(3, n); using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, lim n→∞ t(m,n) r(m,n) = 0 for each fixed m ≥ 3, is true for m ≤ 4. At last, we prove that s(3, 9) and t(3, 9) are both equal to 26.

Journal of Combinatorial Theory, Series B, 1983

The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G. A conjecture of P. Erdos and S. Burr is settled in the afftrmative by proving that for each d > 1, there exists a constant c so that if G is any graph on n vertices with maximum degree d, then the Ramsey number of G is at most cn.

Applied Mathematics Letters, 2004

Discrete Mathematics, 2004

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

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) .

Lecture Notes in Computer Science, 2018

The Ramsey number R X (p, q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey number is linear in X if there is a constant k such that R X (p, q) ≤ k(p + q) for all p, q. In the present paper we conjecture that Ramsey number is linear in X if and only if the co-chromatic number is bounded in X and prove a number of results supporting the conjecture.

Bulletin of the London Mathematical Society, 1974

Continuing this series of papers on generalized Ramsey theory for graphs, we define the Ramsey number r(D lt D 2) of two digraphs D t and D 2 as the minimum p such that every 2-colouring of the arcs (directed lines) of DK P (the complete symmetric digraph of order p) contains a monochromatic D t or D 2. It is shown (Theorem 1) that this number exists if and only if D y or D 2 is acyclic. Then r(D), the diagonal Ramsey number of a given acyclic digraph D, is defined as r(D, D). Notation: D' is the converse of D, GD is the underlying graph of D, DG is the symmetric digraph of G, and T p is the transitive tournament of order p. Let r(m, n) be the traditional Ramsey number of the two complete graphs K m and K n. Finally, let S n be the star with n arcs from one point u to n points v r Assuming the Ramsey numbers under discussion exist, we prove the following results: THEOREM 2. r(D u D 2) = r(D x ', D 2). THEOREM 3. r(D u D 2) ^ r(GD x , GD 2). THEOREM 4. r(D u D 2) < r(T Pl , T P2) if both D t and D 2 (with p x and p 2 points respectively) are acyclic.

Discrete Mathematics, 2001

For each vertex s of the subset S of vertices of a graph G, we deÿne Boolean variables p; q; r which measure the existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p; q; r) may be considered as a compound existence property of S-pns. The set S is called an f-set of G if f = 1 for all s ∈ S and the class of all f-sets of G is denoted by f . Special cases of f include the independent sets, irredundant sets and CO-irredundant sets of G. For some f ∈ F it is possible to deÿne analogues (involving f-sets) of the classical Ramsey graph numbers. We consider existence theorems for these f-Ramsey numbers and prove that some of them satisfy the well-known recurrence inequality which holds for the classical Ramsey numbers.

IAENG International Journal of Applied Mathematics

For positive integers k and l , the Ramsey number R(k,l) is the least positive integer n such that for every graph G of order n , either G contains k K as a subgraph or G contains l K as a subgraph. In this paper it is shown that Ramsey numbers ≥ R(k,l) 2kl -3k -3l + 6 when ≤ ≤ 3 k l , and ≥ R(k,l) 2kl -3k + 2l -12 when ≤ ≤ 5 k l .

Discrete Mathematics, 1982

Graphs and Combinatorics, 2004

A color pattern is a graph whose edges are partitioned into color classes. A family F of color patterns is a Ramsey family if there is some integer N such that every edgecoloring of K N has a copy of some pattern in F. The smallest such N is the (pattern) Ramsey number R(F) of F. The classical Canonical Ramsey Theorem of Erdős and Rado [4] yields an easy characterization of the Ramsey families of color patterns. In this paper we determine R(F) for all families consisting of equipartitioned stars, and we prove that 5 s−1 2 +1 ≤ R(F) ≤ 3s− √ 3s when F consists of a monochromatic star of size s and a polychromatic triangle.

2010

The Ramsey number r(G) of a graph G is the minimum N such that every red-blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most 2 O(√ m). Motivated by this estimate Erdős conjectured, more than a quarter century ago, that there is an absolute constant c such that r(G) ≤ 2 c √ m for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.

COMBINATORICA, 1998

We investigate the induced Ramsey number r ind (G, H) of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with k = |V (G)| ≤ t = |V (H)|, we have