For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is the smallest positive integer N with the property that every graph on at least N vertices contains F1 or its complement contains F2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θn,Km) for m = 2, 3, 4 and n > m. More specifically, we establish that r(θn,Km) = (n− 1)(m− 1) ...

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...

We determine minimal elements, i.e., atoms, in certain partial orders of factor closed languages under ⊆. This is in analogy to structural Ramsey theory which determines minimal structures in partial orders under embedding. Mathematics Subject Classification. 68R15, 05C55.

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number R(4, 3, 3) = 30. The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more ...

Giraud (1968) demonstrated a process for constructing cyclic Ramsey graph colourings, starting from a known cyclic ‘prototype’ colouring, adding edges of a single new colour, and producing a larger cyclic pattern. This paper describes an extension of that construction which allows any number of new colours to be introduced simultaneously, by using two multicolour prototypes, each of which is a ...

Let Kl×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G1 and G2, and integer j ≥ 2, the size multipartite Ramsey number mj(G1, G2) is the smallest integer t such that every factorization of the graph Kj×t := F1 ⊕ F2 satisfies the following condition: either F1 contains G1 or F2 contains G2. In 2007, Syafrizal e...

For arbitrary graphs G1 and G2, define the Ramsey number R(G1,G2) to be the minimum positive integer N such that in every bicoloring of edges of the complete graph KN with, say, red and blue colors, there is either a red copy of G1 or a blue copy of G2. The classical Ramsey number r(k, l) is in our terminology R(Kk,Kl). Call a family F of graphs linear Ramsey if there exists a constant C = C(F)...

The star-critical Ramsey number r∗(H1,H2) is the smallest integer k such that every red/blue coloring of the edges of Kn −K1,n−k−1 contains either a red copy of H1 or a blue copy of H2, where n is the graph Ramsey number R(H1,H2). We study the cases of r∗(C4, Cn) and R(C4,Wn). In particular, we prove that r∗(C4, Cn) = 5 for all n > 4, obtain a general characterization of Ramsey-critical (C4,Wn)...

We determine the value of the Ramsey number R(W5, K5) to be 27, where W5 = K1 + C4 is the 4-spoked wheel of order 5. This solves one of the four remaining open cases in the tables given in 1989 by George R. T. Hendry, which included the Ramsey numbers R(G, H) for all pairs of graphs G and H having five vertices, except seven entries. In addition, we show that there exists a unique up to isomorp...