The flip graph of triangulations has as vertices all triangulations of a convex @math -gon, and an edge between any two triangulations that differ in exactly one edge. An @math -rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly @math times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of @math -rainbow cycles for three different flip graphs on geometric classes of objects: the aforementioned flip graph of triangulations of a convex @math -gon, the flip graph of plane trees on an arbitrary set of @math points, and the flip graph of non-crossing perfect matchings on a set of @math points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of @math and the flip graph of @math -element subsets of @math . In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of @math , @math and @math .
Rainbow cycles and paths have also been studied in graphs other than flip graphs. A well-known conjecture in this context due to Andersen @cite_7 asserts that every properly edge-coloured complete graph on @math vertices has a rainbow path of length @math , i.e., a path that has distinct colors along each of its edges. Progress towards resolving this conjecture was recently made by Alon, Pokrovskiy and Sudakov @cite_11 , and Balogh and Molla @cite_5 .
@cite_5 @cite_7 @cite_11
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A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph @math has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost @math . In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured @math has a rainbow cycle of length @math . One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured @math formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties. We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen. It is well known that every planar graph G is 2-colorable in such a way that no 3-cycle of G is monochromatic. In this paper, we prove that G has a 2-coloring such that no cycle of length 3 or 4 is monochromatic. The complete graph K5 does not admit such a coloring. On the other hand, we extend the result to K5-minor-free graphs. There are planar graphs with the property that each of their 2-colorings has a monochromatic cycle of length 3, 4, or 5. In this sense, our result is best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 25–38, 2004 Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004].