Given a graph G and a coloring of its edges, a subgraph of G is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous Caccetta–Häggkvist conjecture, proposed by the first author, is that if in a coloring of the edge set of an n-vertex graph by n colors, in which each color class is of size k, the rainbow girth is at most \(\left\lceil {{n \over k}} \right\rceil \). In the known examples for sharpness of this conjecture the color classes are stars, suggesting that when the color classes are matchings, the result may be improved. We show that the rainbow girth of n matchings of size at least 2 is O(log n).

给定图G及其边的颜色，如果其边具有不同的颜色，则G的子图称为彩虹。G的边缘着色的彩虹周长是G中彩虹周期的最小长度。第一作者提出的著名的 Caccetta–Häggkvist 猜想的推广是，如果用n 种颜色对n顶点图的边集进行着色，其中每个颜色类的大小为k，则彩虹周长为最多\(\left\lceil {{n \over k}} \right\rceil \)。在该猜想的清晰度的已知示例中，颜色类别是星形，这表明当颜色类别匹配时，结果可能会得到改善。我们证明， n 个大小至少为 2 的匹配的彩虹周长是O (log n )。