Rainbow planar Tur{\'a}n numbers of cycles

TL;DR

This paper investigates the maximum edges in planar graphs with proper edge coloring that avoid rainbow cycles, providing exact values for certain cycle lengths and conditions.

Contribution

It introduces the concept of rainbow planar Turán numbers for cycles and determines exact values for specific cycle lengths and graph sizes.

Findings

Exact value of ${ ext{ex}_{ ext{P}}}^*(n, C_3)$ as 2n-4.

${ ext{ex}_{ ext{P}}}^*(n, C_4)$ equals 3n-6 for specific n.

${ ext{ex}_{ ext{P}}}^*(n, C_k)$ equals 3n-6 for all k ≥ 5 and large enough n.

Abstract

The rainbow Tur{\'a}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{\'a}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{\'a}n number of cycles. Since $C_3$ is complete, ${\text{ex}_{\mathcal{P}}}^*(n, C_3)$ is exactly its planar Tur{\'a}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 $n\ge 3$.