Paths and cycles with many colors in edge-colored complete graphs

In this paper we consider properly edge-colored complete graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A proper edge-coloring is a factorization if each color class is a perfect or near perfect matching. A subgraph is called rainbow if its edges have different colors. We show that in any factorization of the complete graph Kn on n vertices we have a rainbow linear forest with at least (1− o(1))n edges, and thus we also have a Hamiltonian cycle with at least (1− o(1))n distinct colors. Some of our results can be interpreted for Latin squares, providing large partial transversals without short cycles. We also show that in any properly edge-colored Kn we have a rainbow cycle with at least (4/7− o(1))n edges. 1 Paths and cycles with many colors V (G) and E(G) denote the vertex-set and the edge-set of the graph G. Kn is the complete graph on n vertices, where n ≥ 2 is assumed throughout. Cl (Pl) is the cycle (path) with l edges. A linear forest is a vertex disjoint collection of paths. A maxdegree-2 graph is a graph where all the degrees are at most 2, thus the graph is a collection of cycles, paths and isolated points. In this paper we consider properly edge-colored complete graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A proper edge-coloring is a factorization if each color class is a perfect or near perfect matching (so it leaves out only at most one vertex). A subgraph is called rainbow if its edges have different colors. There has been extensive research on paths and cycles with many colors. In [3] it is shown that for every ε > 0, there exists n0(ε) such that for every n ≥ n0(ε), in any edge-coloring of Kn where no vertex is incident with more than ( 1− 1 √ 2 − ε ) n edges of the same color, there is a Hamiltonian cycle in which adjacent edges have distinct colors. An edge-coloring of Kn is called a k-bounded coloring if no color is used more than k = k(n) times. Several researchers studied the question of how fast one can allow k to grow and still guarantee a rainbow Hamiltonian cycle. This problem is mentioned in Erdős, Nesetril and Rödl [5]. There they refer to it as an Erdős-Stein problem and show that k can be any constant. Hahn and Thomassen [9] then showed that k could grow as fast as n and in fact Hahn conjectured (see [9]) that the growth of k could be linear in n. In an unpublished work Rödl and Winkler improved this to n. Frieze and Reed [8] improved this further to n/(c log n) for some constant c (throughout the paper log denotes the natural based logarithm). Finally, using a method of Erdős and Spencer [6], where they introduced the Lopsided Local Lemma, Albert, Frieze and Reed proved the Hahn conjecture by showing that k could be dcne, for any constant c < 1/32 if n ≥ n0(c). See also [7] for related results.