Abstract
AbstractOne of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős‐Rényi random graph Gn,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs.In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014
Highlights
Hamilton cycles occupy a position of central importance in graph theory, and are the subject of countless results
In the context of random structures, much research has been done on many aspects of Hamiltonicity, in a variety of random structures
In this paper we consider the existence of rainbow Hamilton cycles in edge-colored graphs. (A set S of edges is called rainbow if each edge of S has a different color.) There are two general types of results in this area: existence whp1 under random coloring and guaranteed existence under adversarial coloring
Summary
Hamilton cycles occupy a position of central importance in graph theory, and are the subject of countless results. In this paper we consider the existence of rainbow Hamilton cycles in edge-colored graphs. Nesetril, and Rodl [13], Hahn and Thomassen [17] and Albert, Frieze, and Reed [1] (correction in Rue [24]) considered colorings of the edges of the complete graph Kn where no color is used more than k times It was shown in [1] that if k ≤ n/64, there must be a multi-colored Hamilton cycle. A result of Janson and Wormald [16] on rainbow Hamilton cycles in randomly edge-colored random regular. A random 3-uniform hypergraph gives rise to a randomly edge-colored random graph We will discuss this further, when we use the reverse connection to realize one part of our new result. All logarithms will be in base e ≈ 2.718
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
