Rainbow numbers for matchings in plane triangulations

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number of Hwith respect to G. If G is a complete graph Kn, the numbers f(Kn,H) and rb(Kn,H) are called anti-Ramsey numbers and rainbow numbers, respectively. In this paper we will study the existence of rainbow matchings in plane triangulations. Denote by kK2 a matching of size k and Tn the class of all plane triangulations of order n. The rainbow numberrb(Tn,kK2) is the minimum number of colors c such that, if kK2⊆Tn∈Tn, then any edge-coloring of Tn with at least c colors contains a rainbow copy of kK2. We give lower and upper bounds on rb(Tn,kK2) for all k≥3 and n≥2k. Furthermore, we obtain the exact values of rb(Tn,kK2) for 2≤k≤4 and n≥2k.