The asymptotic $χ$-boundedness of hereditary families
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912408927207424 |
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| author | Reed, Bruce Yuditsky, Yelena |
| author_facet | Reed, Bruce Yuditsky, Yelena |
| contents | A family ${\cal F}$ of graphs is asymptotically $χ$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $χ(G) \le f(ω(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $χ$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $χ(G)=ω(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $χ(G) = ω(G)$. We show that the $C_6$-free graphs are asymptotically $χ$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The asymptotic $χ$-boundedness of hereditary families Reed, Bruce Yuditsky, Yelena Combinatorics 05C80 05C15 G.2.2 A family ${\cal F}$ of graphs is asymptotically $χ$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $χ(G) \le f(ω(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $χ$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $χ(G)=ω(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $χ(G) = ω(G)$. We show that the $C_6$-free graphs are asymptotically $χ$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$. |
| title | The asymptotic $χ$-boundedness of hereditary families |
| topic | Combinatorics 05C80 05C15 G.2.2 |
| url | https://arxiv.org/abs/2506.01070 |