On Generalized Ramsey Theory: The Bipartite Case

Maria Axenovich, Zoltán Füredi, Dhruv Mubayi

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

Given graphs G and H, a coloring of E(G) is called an (H, q)-coloring if the edges of every copy of H ⊆ G together receive at least q colors. Let r(G, H, q) denote the minimum number of colors in an (H, q)-coloring of G. We determine, for fixed p, the smallest q for which r(Kn,n, Kp,p, q) is linear in n, the smallest q for which it is quadratic in n. We also determine the smallest q for which r(Kn,n, Kp,p, q) =n2 - O(n), and the smallest q for which r(Kn,n, Kp,p, q) = n2 - O(1). Our results include showing that r(Kn,n, K2, t+1, 2) and r(Kn, K2, t+1, 2) are both (1 + 0(1)) √n/t as n → ∞, thereby proving a special case of a conjecture of Chung and Graham. Finally, we determine the exact value of r(Kn,n, K3,3, 8), and prove that 2n/3≤r(Kn,n,C4, 3)≤n + 1. Several problems remain open.

Original languageEnglish
Pages (from-to)66-86
Number of pages21
JournalJournal of Combinatorial Theory. Series B
Volume79
Issue number1
DOIs
Publication statusPublished - May 2000

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Keywords

  • Algebraic constructions
  • Edge-coloring of bipartite graphs
  • Projective planes
  • Ramsey theory

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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