Vertex-disjoint rainbow cycles in edge-colored graphs

Let G be an edge-colored graph of order n. The color-neighborhood of a vertex u of G is the set of colors of the edges incident with u in G, denoted by CNG(u), or CN(u) for short. A subgraph F of G is called rainbow if any two edges of F have distinct colors. In this paper, we first give a sufficient condition for the existence of rainbow cycles by using color-neighborhood unions of pairs of vertices in G. In 2019, Fujita et al. showed that G contains k vertex-disjoint rainbow cycles if |CN(x) boolean OR & nbsp;CN(y)|& nbsp;>=& nbsp;n/2 + 64k + 1 for any two vertices x, y of G. We obtain a result that G contains k vertex-disjoint rainbow cycles if |CN(x) boolean OR & nbsp;CN(y)|& nbsp;>=& nbsp;n/2 + 18k + 1 for any two vertices x, y of G. Furthermore, we give better bounds for k = 2, 3. Finally, we show that G contains two vertex-disjoint rainbow cycles of different lengths if |CN(x) boolean OR CN(y)| >=& nbsp;2n/3 + 6 for every pair of vertices x, y of G. (C)& nbsp;2022 Elsevier B.V. All rights reserved.