Ramsey-type problems on induced covers and induced partitions toward the Gy\'{a}rf\'{a}s-Sumner conjecture

TL;DR

This paper investigates Ramsey-type problems related to induced covers and partitions in graphs, providing new results that support the Gyárfás-Sumner conjecture and related invariants, advancing understanding of graph coloring and structure.

Contribution

The paper introduces and solves Ramsey-type problems for induced star and path covers and partitions, extending the Gyárfás-Sumner conjecture to these invariants.

Findings

Established bounds for induced SP-cover and SP-partition numbers in T-free graphs.

Proved that these invariants satisfy Ramsey-type properties analogous to the Gyárfás-Sumner conjecture.

Settled several open problems related to induced covers and partitions in graph theory.

Abstract

Gy\'{a}rf\'{a}s 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 $\chi (G)\leq f_{T}(\omega (G))$, where $\chi (G)$ and $\omega (G)$ 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 Gy\'{a}rf\'{a}s-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants.