Chen et al. (Appl Math Lett 17:281–285, 2004) conjectured that for even m, $$R(T_n,W_m)=2n-1$$ if the maximum degree $$\varDelta (T_n)$$ is small. However, they did not state how small it is. Related to this conjecture, also interesting know which tree $$T_n$$ causes Ramsey number $$R(T_n,W_m)$$ be greater than $$2n-1$$ whenever m even. In paper, we determine $$R(T_n,W_8)$$ all trees with (T_n)...

For given graphs G and H; the Ramsey number R(G;H) is the leastnatural number n such that for every graph F of order n the followingcondition holds: either F contains G or the complement of F contains H.In this paper firstly, we determine Ramsey number for union of pathswith respect to sunflower graphs, For m ≥ 3, the sunflower graph SFmis a graph on 2m + 1 vertices obtained...

The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω. We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In thi...

Let L be a disjoint union of nontrivial paths. Such a graph we call a linear forest. We study the relation between the 2-local Ramsey number R2-loc(L) and the Ramsey number R(L), where L is a linear forest. L will be called an (n, j)-linear forest if L has n vertices and j maximal paths having an odd number of vertices. If L is an (n, j)-linear forest, then R2-loc(L) = (3n − j)/2 + dj/2e −

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers R(m,n) with m, n≥3, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers R(m,n). We show how the computation of R(m,n) can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evol...

Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X] of unordered n-tuples from X, find a subset Y ⊆ X such that all n-tuples in [Y ] get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is l...

It is shown that if G and H are star-forests with no single edge stars, then (G, H) is Ramsey-finite if and only if both G and H are single stars with an odd number of edges . Further (5,,, U kS 1 , S, U 1S T ) is Ramsey-finite when m and n are odd, where S, denotes a star with i edges . In general, for G and H star-forests, (G U kS i , H U lS,) can be shown to be Ramsey-finite or Ramsey-infini...

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 work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of ...