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; DOI: 10.1006/aama.1998.0620

ABSTRACT Giveni, jpositive integers, letKi, jdenote a bipartite complete graph and letRr(m, n) be the smallest integerasuch that for anyr-coloring of the edges ofKa, aone can always find a monochromatic subgraph isomorphic toKm, n. In other words, ifa ≥ Rr(m, n) then every matrixa × awith entries in {0, 1,…,r − 1} always contains a submatrixm × norn × mwhose entries arei, 0 ≤ i ≤ r − 1. We shall prove thatR2(m, n) ≤ 2m(n − 1) + 2m − 1 − 1, which generalizes the previous resultsR2(2, n) ≤ 4n − 3 andR2(3, n) ≤ 8n − 5 due to Beineke and Schwenk. Moreover, we find a class of lower bounds based on properties of orthogonal Latin squares which establishes that limr → ∞ Rr(2, 2)r − 2 = 1.

0 Bookmarks
 · 
55 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we establish a connection between polarized partition relations and the functionAq (n, d) of coding theory, which implies some known results and new examples of polarized relations. A connection between Aq(n, d) and the Zarankiewicz numbers is also discussed.
    European Journal of Combinatorics 03/2001; 22:351-356. DOI:10.1006/eujc.2000.0474 · 0.61 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    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 vertex-disjoint subgraphs isomorphic to H. For every graph class F, let F-Subgraph Test and F-Packing 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) polynomial-time solvable, * admits a polynomial (many-one) kernel, or * admits a polynomial Turing kernel (that is, has an adaptive polynomial-time 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 F-Packing admits a polynomial kernel, and has no polynomial (many-one) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if F does not have this property, then F-Packing is either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does not admit polynomial Turing kernels either. For F-Subgraph 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 F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard 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 F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard, W[1]-hard, or Long Path-hard, otherwise.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Some bounds for G 1 -G 2 bipartite Ramsey numbers b(G 1 ;G 2 ) are given, which imply that b(K 2,2 ;K 1,n )=n+q for the range q 2 -q+1≤n≤q 2 , where q is a prime power. Our new construction establishes in particular that b(K 2,n ;K 2,n )=4n-3 if 4n-3 is a prime power, reinforcing a weaker form of a conjecture due to Beineke and Schwenk. Particular relationships between b(G 1 ;G 2 ) and G 1 -G 2 Ramsey numbers are also determined.
    Discrete Mathematics 08/2000; 223(1). DOI:10.1016/S0012-365X(00)00041-8 · 0.57 Impact Factor

Full-text (2 Sources)

Download
50 Downloads
Available from
May 29, 2014