Reuniting $χ$-boundedness with polynomial $χ$-boundedness
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917204208910336 |
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| author | Chudnovsky, Maria Cook, Linda Davies, James Oum, Sang-il |
| author_facet | Chudnovsky, Maria Cook, Linda Davies, James Oum, Sang-il |
| contents | A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11167 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reuniting $χ$-boundedness with polynomial $χ$-boundedness Chudnovsky, Maria Cook, Linda Davies, James Oum, Sang-il Combinatorics Discrete Mathematics 05C15 G.2.2 A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna. |
| title | Reuniting $χ$-boundedness with polynomial $χ$-boundedness |
| topic | Combinatorics Discrete Mathematics 05C15 G.2.2 |
| url | https://arxiv.org/abs/2310.11167 |