Improved bounds for rainbow numbers of matchings in plane triangulations

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ 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 $kK_2$ a matching of size $k$ and $\mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. In [S. Jendrol$'$, I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158--164], the authors determined the exact values of $rb(\mathcal {T}_n, kK_2)$ for $2\leq k \le 4$ and proved that $2n+2k-9 \le rb(\mathcal {T}_n, kK_2) \le 2n+2k-7+2\binom{2k-2}{3}$ for $k \ge 5$. In this paper, we improve the upper bounds and prove that $rb(\mathcal {T}_n, kK_2)\le 2n+6k-16$ for $n \ge 2k$ and $k\ge 5$. Especially, we show that $rb(\mathcal {T}_n, 5K_2)=2n+1$ for $n \ge 11$.