Rainbow planar Tur{á}n numbers of cycles
Xiaonan Liu
TL;DR
The paper addresses the rainbow planar Turán problem for cycles, defining $\text{ex}_{\mathcal{P}}^*(n,H)$ as the maximum edge count in an $n$-vertex planar graph that can be properly edge-colored without a rainbow copy of $H$. Using explicit planar triangulations with carefully designed edge-colorings, the authors derive tight bounds and constructions, notably showing $\text{ex}_{\mathcal{P}}^*(n, C_3)=2n-4$, $\text{ex}_{\mathcal{P}}^*(n, C_4)=3n-6$ for specific $n$ and all $k\ge 5$, and $\text{ex}_{\mathcal{P}}^*(n, C_k)=3n-6$ for all $k\ge 5$, $n\ge 3$. For $C_5$ and $C_6$, they provide $F_n$-based triangulations with no rainbow copies, establishing $\text{ex}_{\mathcal{P}}^*(n, C_5)=\text{ex}_{\mathcal{P}}^*(n, C_6)=3n-6$ for all $n\ge 3$. These results extend rainbow Turán theory to planar graphs, offering exact values in several regimes and illuminating the interplay between triangulations, colorings, and rainbow cycle avoidance.
Abstract
The rainbow Tur{á}n number of a fixed graph $H$, denoted by ${\text{ex}}^*(n,H)$, is the maximum number of edges in an $n$-vertex graph such that it admits a proper edge coloring with no rainbow $H$. We study this problem in planar setting. The rainbow planar Tur{á}n number of a graph $H$, denoted by ${\text{ex}_{\mathcal{P}}}^*(n,H)$, is the maximum number of edges in an $n$-vertex planar graph such that it has a proper edge coloring with no rainbow $H$. We consider the rainbow planar Tur{á}n number of cycles. Since $C_3$ is complete, ${\text{ex}_{\mathcal{P}}}^*(n, C_3)$ is exactly its planar Tur{á}n number, which is $2n-4$ for $n\ge 3$. We show that ${\text{ex}_{\mathcal{P}}}^*(n, C_4)=3n-6$ for $n=k^2-3k+2$ where $k\ge 5$, and ${\text{ex}_{\mathcal{P}}}^*(n,C_k)=3n-6$ for all $k\ge 5$ and $n\ge 3$.