Long rainbow cycles in proper edge-colorings of complete graphs

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4/7− o(1))n edges. This improves the lower bound of n/2− 1 proved in [1]. We consider properly edge-colored complete graphs Kn, where two edges with the same color cannot be incident to each other, so each color class is a matching. An important and well investigated special case of proper edge-colorings is a factorization where each color class forms a perfect (if n is even) or nearly perfect (if n is odd) matching. A colored subgraph ofKn is called rainbow if its edges have different colors. The size of rainbow subgraphs of maximum degree two, i.e. union of paths and cycles in proper colorings are well investigated. A consequence of Ryser’s well-known conjecture ([12] stating that every Latin square has a transversal) would be that for odd n in every factorization of Kn there is a rainbow 2-factor (and for even n a 2factor covering all but one vertices). Although this is not known, there were several results that made advances towards Ryser’s conjecture and show the existence of a 2-factor covering n − o(n) vertices, [4, 10, 13, 14]. Andersen [3] applied the method of [4] to prove that in every proper coloring of Kn there is a rainbow subgraph with at least n− √ 2n vertices whose components are paths. Another line of research looked for rainbow Hamiltonian cycles from the assumption that there is an upper bound k on the number of colors in each color class. This problem is mentioned in Erdős, Nesetril and Rödl [5]. Hahn and Thomassen [9] 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. After further improvements [7], Albert, Frieze and Reed [2] proved the Hahn Conjecture by showing that k could be ⌈cn⌉, for any constant c < 1/32 if n ≥ n0(c). See also [6] for related results. Although it is widely believed that in every proper coloring ofKn there is a rainbow path and cycle with length almost n (the obstacle to a spanning rainbow path or cycle comes from a special factorization, see [1], [9], [11]), the above mentioned results do not imply such a bound. As far as we know the best lower bounds are 2n/3 for the path ([8]) and n/2−1 for the cycle ([1]). The purpose of this note is the improvement of the latter result to (1− o(1)) 7 . Theorem 1. For arbitrary ε, where 1/2 > ε > 0, there exists an n0(ε) such that if n ≥ n0(ε), then in any proper edge-coloring of Kn there is a rainbow cycle with length at least ( 4 7 − ε ) n. Proof: The vertex-set and the edge-set of a graph G are denoted by V (G) and E(G). Cl is the cycle with l vertices and Pl is the path with l vertices.