A directed Andrásfai-Erdős-Sós theorem and chromatic profiles of oriented cycles
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911144935948288 |
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| author | Xue, Yisai |
| author_facet | Xue, Yisai |
| contents | The chromatic profile of a digraph $H$, denoted by $δ_χ^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree $δ^{+}(D) \ge dn$ must be $k$-colorable. We determine the exact chromatic profile for several fundamental classes of digraphs. Our main result is a directed analogue of the Andrásfai-Erdős-Sós theorem, stating that $δ_χ^{+}(T_r, r-1)=\frac{3 r-7}{3 r-4}$, where $T_r$ is the transitive tournament on $r$ vertices. We then determine the chromatic profile for directed odd cycles, showing that $δ^+_χ(\overrightarrow{C}_{2\ell+1},2)=1/2$ for all $\ell\ge 1$. Finally, we resolve the profile for the three remaining orientations of the pentagon, establishing that $δ_χ^{+}(C_{5}',2)=δ_χ^{+}(C_{5}'',2)=δ_χ^{+}(C_{5}''',2)=1/3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A directed Andrásfai-Erdős-Sós theorem and chromatic profiles of oriented cycles Xue, Yisai Combinatorics The chromatic profile of a digraph $H$, denoted by $δ_χ^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree $δ^{+}(D) \ge dn$ must be $k$-colorable. We determine the exact chromatic profile for several fundamental classes of digraphs. Our main result is a directed analogue of the Andrásfai-Erdős-Sós theorem, stating that $δ_χ^{+}(T_r, r-1)=\frac{3 r-7}{3 r-4}$, where $T_r$ is the transitive tournament on $r$ vertices. We then determine the chromatic profile for directed odd cycles, showing that $δ^+_χ(\overrightarrow{C}_{2\ell+1},2)=1/2$ for all $\ell\ge 1$. Finally, we resolve the profile for the three remaining orientations of the pentagon, establishing that $δ_χ^{+}(C_{5}',2)=δ_χ^{+}(C_{5}'',2)=δ_χ^{+}(C_{5}''',2)=1/3$. |
| title | A directed Andrásfai-Erdős-Sós theorem and chromatic profiles of oriented cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.07760 |