Rainbow Turán numbers for short brooms
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915166846713856 |
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| author | Byrne, John Gamlath, E. G. K. M Halfpap, Anastasia Miyasaki, Sydney Parker, Alex |
| author_facet | Byrne, John Gamlath, E. G. K. M Halfpap, Anastasia Miyasaki, Sydney Parker, Alex |
| contents | 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 $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rainbow Turán numbers for short brooms Byrne, John Gamlath, E. G. K. M Halfpap, Anastasia Miyasaki, Sydney Parker, Alex Combinatorics 05C35 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 $t$. |
| title | Rainbow Turán numbers for short brooms |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2502.16057 |