Algorithmica https://doi.org/10.1007/s00453-019-00618-0 Quadratic Vertex Kernel for Rainbow Matching Sushmita Gupta 1 · Sanjukta Roy 2 · Saket Saurabh 2,3 · Meirav Zehavi 4 Received: 7 July 2018 / Accepted: 10 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract In this paper, we study the NP-complete colorful variant of the classic matching prob- lem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k , the goal is to decide whether there exists a matching of size at least k such that the edges in the matching have distinct colors. Previously, in [MFCS’17], we studied this problem from the view point of Parameterized Complexity and gave efficient FPT algorithms as well as a quadratic kernel on paths. In this paper we design a quadratic vertex kernel for Rainbow Matching on general graphs; generalizing the earlier quadratic kernel on paths to general graphs. For our kernelization algorithm we combine a graph decomposition method with an application of expansion lemma. Keywords Rainbow matching · Parameterized complexity · Polynomial kernel 1 Introduction The classical notion of matching or matching with constraints have been extensively studied for several decades in the area of Combinatorial Optimization [6,18]. Given an undirected graph G, a set of edges is called a matching if the edges are pairwise non- adjacent. That is, no two edges share a common vertex. In the Maximum Matching B Sanjukta Roy sanjukta@imsc.res.in Sushmita Gupta sushmitagupta@niser.ac.in Saket Saurabh saket@imsc.res.in Meirav Zehavi meiravze@bgu.ac.il 1 National Institute for Science Education and Research, Bhubaneswar, India 2 The Institute of Mathematical Sciences, HBNI, Chennai, India 3 University of Bergen, Bergen, Norway 4 Ben-Gurion University, Beersheba, Israel 123