A note on the rainbow Tur\'an number of brooms with length 2 handles

TL;DR

This paper investigates the maximum edges in large graphs that can be properly edge-colored without containing a rainbow broom with a length 2 handle, correcting previous claims and characterizing extremal structures.

Contribution

It corrects an error in prior work on rainbow Turán numbers of certain broom trees and provides a detailed characterization of extremal graphs for these parameters.

Findings

Corrected the asymptotic value of $ ext{ex}^*(n,B_{k,2})$ for all but two small cases.

Identified specific extremal constructions for certain congruence classes of n modulo k.

Validated the original claims in most cases, with precise exceptions.

Abstract

For a fixed graph $F$, the rainbow Tur\'an number $\mathrm{ex^*}(n,F)$ is the largest number of edges possible in an $n$-vertex graph which admits a rainbow-$F$-free proper edge-coloring. We focus on the rainbow Tur\'an numbers of trees obtained by appending some number of pendant edges to one end of a length 2 path; we call such a tree with $k$ total edges a $k$-edge broom with length $2$ handle, denoted by $B_{k,2}$. Study of $\mathrm{ex^*}(n,B_{k,2})$ was initiated by Johnston and Rombach, who claimed a proof asymptotically establishing the value of $\mathrm{ex^*}(n,B_{k,2})$ for all $k$. We correct an error in this original argument, identifying two small cases in which the value claimed in the literature is incorrect; in all other cases, we recover the originally claimed value. Our argument also characterizes the extremal constructions for $\mathrm{ex^*}(n,B_{k,2})$ for certain congruence classes of $n$ modulo $k$.