Improved bounds for rainbow numbers of matchings in plane triangulations

TL;DR

This paper improves upper bounds for rainbow numbers of matchings in plane triangulations, providing exact values for certain cases and tighter bounds for larger matchings.

Contribution

The authors establish improved upper bounds for rainbow numbers in plane triangulations and determine exact values for specific matching sizes.

Findings

Improved upper bound: rb(_n, kK_2)  2n+6k-16 for n  2k and k  5.

Exact value: rb(_n, 5K_2) = 2n+1 for n  11.

Extended results refine previous bounds for rainbow numbers in plane triangulations.

Abstract

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 $n \ge 11$.