Lower bounds for rainbow Tur\'{a}n numbers of paths and other trees

TL;DR

This paper establishes lower bounds for the maximum number of edges in edge-colored graphs avoiding rainbow paths and certain trees, extending previous results to new classes of graphs.

Contribution

It provides new lower bounds for rainbow Turán numbers for paths and specific trees, generalizing earlier work and covering additional graph classes.

Findings

Lower bound for rainbow Turán number of paths: x^*(n, P_k) rac{k}{2}n + O(1)

Bounds for brooms with 2^s-1 edges and diameter 0

Results for small diameter caterpillars

Abstract

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Tur\'{a}n number of $F$. We show that $ex^*(n,P_k)\geq \frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\leq 10$ and a few other caterpillars of small diameter.