Planar anti-Ramsey numbers for paths and cycles

Motivated by anti-Ramsey numbers introduced by Erd\H{o}s, Simonovits and S'os in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. 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)$suchthat$T$containsnorainbowcopyof$H$. Analogous to anti-Ramsey numbers and Tur'an numbers, planar anti-Ramsey numbers are closely related to planar Tur'an numbers, where the planar Tur'an number of $H$ is the maximum number of edges of a planar graph on $n$ vertices without containing $H$ as a subgraph. The study of $a{r}_{{}_{\mathcal{P}}}(n,H)$ (under the name of rainbow numbers) was initiated by Hor\v{n}'ak, Jendrol${}^{\mathrm{\prime }}$, Schiermeyer and Sot'ak [J Graph Theory 78 (2015) 248–257]. In this paper we study planar anti-Ramsey numbers for paths and cycles. We first establish lower bounds for $a{r}_{{}_{\mathcal{P}}}(n,P_k)$ when $n\ge k\ge 8$. We then improve the existing lower bound for $a{r}_{{}_{\mathcal{P}}}(n,C_k)$ when $k\ge 5$ and $n\ge k^2-k$. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for $a{r}_{{}_{\mathcal{P}}}(n,C_6)$ when $n\ge 8$ and $ar{_$\mathcal{P}$}(n, C_7)$when$n\geq 13$, respectively.