Improved bounds for anti-Ramsey numbers of matchings in outer-planar graphs

Let O-n be the set of all maximal outer-planar graphs of order n . Let ar(O-n, F ) denote the maximum positive integer k such that there is a k-edge-coloring of a graph T in the family O-n which has no rainbow subgraph F . Denote by M-k a matching of size k . In this paper, we prove that ar(O-n, M-k) <= n + 4k- 9 for n >= 3 k - 3 , which expressively improves the existing upper bound for ar(O-n,M-k). We also prove that ar(O-n,M-5) = n + 4 for all n >= 15 . (c) 2021 Elsevier Inc. All rights reserved.