Improved bounds for rainbow numbers of matchings in plane triangulations

Given two graphs G and H, the rainbow numberrb(G,H) for H with respect to G is defined as the minimum number k such that any k-edge-coloring of G contains a rainbow H, i.e., a copy of H, all of whose edges have different colors. Denote by kK2 a matching of size k and Tn the class of all plane triangulations of order n, respectively. In Jendrol′ et al. (2014), the authors determined the exact values of rb(Tn,kK2) for 2≤k≤4 and proved that 2n+2k−9≤rb(Tn,kK2)≤2n+2k−7+22k−23 for k≥5. In this paper, we improve the upper bounds and prove that rb(Tn,kK2)≤2n+6k−16 for n≥2k and k≥5. Especially, we show that rb(Tn,5K2)=2n+1 for n≥11.