Rainbow Tur\'an numbers for short brooms

TL;DR

This paper investigates the maximum edges in large graphs that avoid rainbow copies of certain broom trees with a length-3 handle, providing bounds and constructions that depend on the divisibility of the number of edges.

Contribution

It establishes new bounds on rainbow Turán numbers for broom trees with a length-3 handle and shows their dependence on the divisibility properties of the number of edges.

Findings

Improved upper bounds on rainbow Turán numbers for broom trees with length-3 handles.

Construction of graphs asymptotically matching these bounds for specific cases.

Demonstration of the influence of divisibility properties of the number of edges on these bounds.

Abstract

A graph $G$ is rainbow-$F$-free if it admits a proper edge-coloring without a rainbow copy of $F$. The rainbow Tur\'an number of $F$, denoted $\mathrm{ex^*}(n,F)$, is the maximum number of edges in a rainbow-$F$-free graph on $n$ vertices. We determine bounds on the rainbow Tur\'an numbers of stars with a single edge subdivided twice; we call such a tree with $t$ total edges a $t$-edge \textit{broom} with length-$3$ handle, denoted by $B_{t,3}$. We improve the best known upper bounds on $\mathrm{ex^*}(n,B_{t,3})$ in all cases where $t \neq 2^s - 2$. Moreover, in the case where $t$ is odd and in a few cases when $t \equiv 0 \mod 4$, we provide constructions asymptotically achieving these upper bounds. Our results also demonstrate a dependence of $\mathrm{ex^*}(n,B_{t,3})$ on divisibility properties of $t$.