Unavoidable subgraphs of colored graphs
2008 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 308, no 19, 4396-4413 p.Article in journal (Refereed) PublishedText
A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper. we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let y(k) be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K-k/2's or simply one K-k/2. Bollobas conjectured that for all k and epsilon > 0, there exists an n(k, epsilon) such that if n >= n(k, epsilon) then every two-edge-coloring of K-n, in which the density of each color is at least epsilon, contains a member of this family. We solve this conjecture and present a series of results bounding it (k, s) for different ranges of epsilon. In particular, if epsilon is sufficiently close to 1/2, the gap between out upper and lower bounds for n(k, epsilon) is smaller than those for the classical Ramsey number R(k, k).
Place, publisher, year, edition, pages
Amsterdam: Elsevier, 2008. Vol. 308, no 19, 4396-4413 p.
Ramsey theory, unavoidable structures
IdentifiersURN: urn:nbn:se:umu:diva-117479DOI: 10.1016/j.disc.2007.08.102ISI: 000257978700011OAI: oai:DiVA.org:umu-117479DiVA: diva2:911396
Workshop on Extremal Graph Theory, JUN 23-27, 2003, Csopak, HUNGARY