Rainbow Turán numbers for short brooms
John Byrne, E. G. K. M Gamlath, Anastasia Halfpap, Sydney Miyasaki, Alex Parker
TL;DR
The paper studies rainbow-Turán numbers ex^*(n,F) for broom trees B_{t,3}, focusing on the dependence of ex^*(n,B_{t,3}) on the parity and divisibility of t. It establishes the sharp asymptotic for odd t: ex^*(n,B_{t,3}) = \\frac{t}{2}n + O(1), and derives nuanced upper bounds for even t that hinge on whether t+2 is a power of two. For t = 2^s and t = 3^s - 1, it provides explicit algebraic colorings that asymptotically achieve the lower bound, while for t ≡ 0 \\pmod{4} it proves a tighter upper bound of ex^*(n,B_{t,3}) \\le \\frac{t}{2}n + O(1). The work also analyzes a concrete unresolved case, t = 10, and demonstrates the non-uniform extremal landscape of rainbow brooms, highlighting how divisibility properties influence the extremal structure and potential constructions.
Abstract
A graph $G$ is rainbow-$F$-free if it admits a proper edge-coloring without a rainbow copy of $F$. The rainbow Turán 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 Turán 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 $t$.