Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture

Abstract

Gyárfás and Sumner independently conjectured that for every tree $T$, there exists a function $f_T:N\to N$ such that every $T$-free graph $G$ satisfies $\chi \left(G\right)\le f_T\left(\omega \left(G\right)\right)$, where $\chi \left(G\right)$ and $\omega \left(G\right)$ are the chromatic number and the clique number of $G$, respectively. This conjecture gives a solution of a Ramsey-type problem on the chromatic number. For a graph $G$, the induced SP-cover number $\text{inspc}\left(G\right)$ (resp. the induced SP-partition number $\text{inspp}\left(G\right)$) of $G$ is the minimum cardinality of a family $P$ of induced subgraphs of $G$ such that each element of $P$ is a star or a path and ${\bigcup }_{P\in P}V\left(P\right)=V\left(G\right)$ (resp. ${\dot{\bigcup }}_{P\in P}V\left(P\right)=V\left(G\right)$). 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 $\text{inspc}$ and $\text{inspp}$, 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.