A note on the rainbow Turán number of brooms with length 2 handles
Anastasia Halfpap
TL;DR
This note analyzes rainbow-Turán numbers for broom graphs with a length-2 handle, correcting an error in prior work and delivering a complete parity-based characterization of ex^*(n,B_{k,2}). The authors derive an universal upper bound and then show sharpness by constructing extremal examples that differ for odd and even k, with Plantholt's theorem playing a key role for even k. They also discuss divisibility-driven nuances in extremal constructions and identify when specific constructions achieve the bound exactly. Overall, the work clarifies the asymptotic behavior of ex^*(n,B_{k,2}) and sharpens our understanding of rainbow-F-free colorings in trees.
Abstract
For a fixed graph $F$, the rainbow Turán 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 Turán 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 $k$.