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arXiv · 2502.16057

Rainbow Turán numbers for short brooms

John Byrne·E. G. K. M Gamlath·Anastasia Halfpap·Sydney Miyasaki·Alex Parker
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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 $t$.

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John Byrne, E. G. K. M Gamlath, Anastasia Halfpap, Sydney Miyasaki, Alex Parker. 2025-02-22. Rainbow Turán numbers for short brooms. https://arxiv.org/abs/2502.16057

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