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 r b ( 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 r b ( G , H ) is called the rainbow number of H with respect to G . If G is a complete graph K n , the numbers f ( K n , H ) and r b ( K n , H ) are called anti-Ramsey numbers and rainbow numbers, respectively. s paper we will study the existence of rainbow matchings in plane triangulations. Denote by k K 2 a matching of size k and T n the class of all plane triangulations of order n . The rainbow number r b ( T n , k K 2 ) is the minimum number of colors c such that, if k K 2 ⊆ T n ∈ T n , then any edge-coloring of T n with at least c colors contains a rainbow copy of k K 2 . We give lower and upper bounds on r b ( T n , k K 2 ) for all k ≥ 3 and n ≥ 2 k . Furthermore, we obtain the exact values of r b ( T n , k K 2 ) for 2 ≤ k ≤ 4 and n ≥ 2 k .