The Outer-Planar Anti-Ramsey Number of Matchings

Abstract

A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let ar(G,H) denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by kK2 a matching of size k and On the class of all maximal outer-planar graphs on n vertices, respectively. The outer-planar anti-Ramsey number of graph H, denoted by ar(On,H), is defined as max{ar(On,H)|On∈On}. It seems nontrivial to determine the exact values for ar(On,H) because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that ar(On,kK2)≤n+3k−8 for all n≥2k and k≥6, which improves the existing upper bound for ar(On,kK2), and prove that ar(On,kK2)=n+2k−5 for n=2k and k≥5. We also obtain that ar(On,6K2)=n+6 for all n≥29.