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

摘要

Let On be the set of all maximal outer-planar graphs of order n. Let ar(On,F) denote the maximum positive integer k such that there is a k-edge-coloring of a graph T in the family On which has no rainbow subgraph F. Denote by Mk a matching of size k. In this paper, we prove that ar(On,Mk)≤n+4k−9 for n≥3k−3, which expressively improves the existing upper bound for ar(On,Mk). We also prove that ar(On,M5)=n+4 for all n≥15.