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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2205.14466 |
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| _version_ | 1866913315998924800 |
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| author | Chiba, Shuya Furuya, Michitaka |
| author_facet | Chiba, Shuya Furuya, Michitaka |
| contents | Gyárfás and Sumner independently conjectured that for every tree $T$, there exists a function $f_{T}:\mathbb{N}\rightarrow \mathbb{N}$ such that every $T$-free graph $G$ satisfies $χ(G)\leq f_{T}(ω(G))$, where $χ(G)$ and $ω(G)$ are the {\it chromatic number} and the {\it clique number} of $G$, respectively. This conjecture gives a solution of a Ramsey-type problem on the chromatic number.
For a graph $G$, the {\it induced SP-cover number ${\rm inspc}(G)$} (resp. the {\it induced SP-partition number ${\rm inspp}(G)$}) of $G$ is the minimum cardinality of a family $\mathcal{P}$ of induced subgraphs of $G$ such that each element of $\mathcal{P}$ is a star or a path and $\bigcup _{P\in \mathcal{P}}V(P)=V(G)$ (resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(G)$). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey-type problems for two invariants ${\rm inspc}$ and ${\rm inspp}$, which are analogies of the Gyárfás-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_14466 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ramsey-type problems on induced covers and induced partitions toward the Gyárfás-Sumner conjecture Chiba, Shuya Furuya, Michitaka Combinatorics Gyárfás and Sumner independently conjectured that for every tree $T$, there exists a function $f_{T}:\mathbb{N}\rightarrow \mathbb{N}$ such that every $T$-free graph $G$ satisfies $χ(G)\leq f_{T}(ω(G))$, where $χ(G)$ and $ω(G)$ are the {\it chromatic number} and the {\it clique number} of $G$, respectively. This conjecture gives a solution of a Ramsey-type problem on the chromatic number. For a graph $G$, the {\it induced SP-cover number ${\rm inspc}(G)$} (resp. the {\it induced SP-partition number ${\rm inspp}(G)$}) of $G$ is the minimum cardinality of a family $\mathcal{P}$ of induced subgraphs of $G$ such that each element of $\mathcal{P}$ is a star or a path and $\bigcup _{P\in \mathcal{P}}V(P)=V(G)$ (resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(G)$). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey-type problems for two invariants ${\rm inspc}$ and ${\rm inspp}$, which are analogies of the Gyárfás-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants. |
| title | Ramsey-type problems on induced covers and induced partitions toward the Gyárfás-Sumner conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2205.14466 |