A note on the rainbow Turán number of brooms with length 2 handles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909968464084992 |
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| author | Halfpap, Anastasia |
| author_facet | Halfpap, Anastasia |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the rainbow Turán number of brooms with length 2 handles Halfpap, Anastasia Combinatorics 05C35 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 $k$. |
| title | A note on the rainbow Turán number of brooms with length 2 handles |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2512.15978 |