Planar anti-Ramsey numbers of matchings

Abstract

Given a positive integer $n$ and a planar graph $H$, let ${\mathcal{T}}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $a{r}_{{}_{\mathcal{P}}}(n,H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in {\mathcal{T}}_n(H)$ such that $T$ contains no rainbow copy of $H$. In this paper we study planar anti-Ramsey numbers of matchings. For all $t\ge 1$, let $M_t$ denote a matching of size $t$. We prove that for all $t\ge 6$ and $n\ge 3t-6$, $2n+3t-15\le a{r}_{{}_{\mathcal{P}}}(n,M_t)\le 2n+4t-14$, which significantly improves the existing lower and upper bounds for $a{r}_{{}_{\mathcal{P}}}(n,M_t)$. It seems that for each $t\ge 6$, the lower bound we obtained is the exact value of $a{r}_{{}_{\mathcal{P}}}(n,M_t)$ for sufficiently large $n$. This is indeed the case for $M_6$. We prove that $a{r}_{{}_{\mathcal{P}}}(n,M_6)=2n+3$ for all $n\ge 30$.