Article
On the Ramsey Problem for Multicolor Bipartite Graphs
Department of Mathematics, State University of Maringá, Maringá, PR, Brazil
Advances in Applied Mathematics
(Impact Factor: 0.88).
01/1999;
22(1):4859.
DOI: 10.1006/aama.1998.0620
Fulltext
Walter Carnielli, May 29, 2015 Available from:
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ABSTRACT: We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, Packing asks if G contains t vertexdisjoint subgraphs isomorphic to H. For every graph class F, let FSubgraph Test and FPacking be the special cases of the two problems where H is restricted to be in F. Our goal is to study which classes F make the two problems tractable in one of the following senses: * (randomized) polynomialtime solvable, * admits a polynomial (manyone) kernel, or * admits a polynomial Turing kernel (that is, has an adaptive polynomialtime procedure that reduces the problem to a polynomial number of instances, each of which has size bounded polynomially by the size of the solution). We identify a simple combinatorial property such that if a hereditary class F has this property, then FPacking admits a polynomial kernel, and has no polynomial (manyone) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if F does not have this property, then FPacking is either WK[1]hard, W[1]hard, or Long Pathhard, giving evidence that it does not admit polynomial Turing kernels either. For FSubgraph Test, we show that if every graph of a hereditary class F satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then FSubgraph Test is solvable in randomized polynomial time and it is NPhard otherwise. We introduce a combinatorial property called (a,b,c,d)splittability and show that if every graph in a hereditary class F has this property, then FSubgraph Test admits a polynomial Turing kernel and it is WK[1]hard, W[1]hard, or Long Pathhard, otherwise. 
Article: Degree Ramsey Numbers of Graphs
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ABSTRACT: Let H G mean that every scolouring of E(H) produces a monochromatic copy of G in some colour class. Let the scolour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): H G}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the doublestar Sa,b, the tree whose two nonleaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edgecolourings of boundeddegree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2edgecolouring such that the subgraph in each colour class has girth at least 5.Combinatorics Probability and Computing 03/2012; 21(12). DOI:10.1017/S0963548311000617 · 0.62 Impact Factor 
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