Bounds for the disjoint unions theorem

Canonization for Disjoint Unions of Theories

If there exist efficient procedures (canonizers) for reducing terms of two first-order theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) cannot be combine...

Canonization for Disjoint Unions of Theories

The Complexity of Unions of Disjoint Sets

This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions o...

The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel Wm or a complete graph Km. We show that if n ≥ 3, then R(kSn, W4) = (k+1)n for k ≥ 2, even n and R(kSn,W4) = (k+1)n−1 for k ≥ 1...

The Ramsey numbers for disjoint unions of cycles

As usual, for simple graphs G and H, let the Ramsey number r(G,H) be defined as the least number n such that for any graph K of order n, either G is a subgraph of K or H is a subgraph of / ( . We shall establish the values of r(aC~,bCs) and r(aCv, bC7) almost precisely (where nG is the graph consisting of n vertex disjoint copies of G) extending the work of Mizuno and Sato, who proved similar r...