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 $ar_{_\mathcal{P}}(n, H)$, 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 $H$. 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 $ar_{_\mathcal{P}}(n, H)$ (under the name of rainbow numbers) was initiated by Hor\v{n}\'ak, Jendrol$'$, 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 $ar_{_\mathcal{P}}(n, P_k)$ when $n\ge k\ge8$. We then improve the existing lower bound for $ar_{_\mathcal{P}}(n, C_k)$ when $k\geq 5$ and $n\geq k^2-k$. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for $ar_{_\mathcal{P}}(n, C_6)$ when $n\ge8$ and $ar_{_\mathcal{P}}(n, C_7)$ when $n\geq 13$, respectively.