On the rainbow planar Tur\'an number of paths

TL;DR

This paper studies the maximum number of edges in planar, properly edge-colored graphs on n vertices that do not contain a rainbow path of length 5, extending the rainbow Turán problem to planar graphs.

Contribution

It determines the exact extremal number for rainbow-avoiding 5-vertex paths in planar graphs, a new result in the variation of the rainbow Turán problem.

Findings

Exact value of $ ext{ex}_{ ext{p}}^*(n, P_5)$ is established.

Provides insights into rainbow path avoidance in planar graphs.

Extends the understanding of rainbow Turán problems to planar graph constraints.

Abstract

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Tur\'an problem, was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path.