Sparse Partitions of Graphs with Bounded Clique Number

TL;DR

The paper proves that graphs with bounded clique number can be partitioned into a limited number of parts with low maximum degree relative to their size, answering a question about sparse partitions.

Contribution

It establishes a bound on the number of parts needed for sparse partitions in graphs with bounded clique number, extending previous results to a broader class of graphs.

Findings

Partition size is at most (1/ε)^{C_r} for graphs with clique number r.

Each part has maximum degree at most ε times its size.

The result generalizes earlier work on graphs excluding induced P_4.

Abstract

We prove that for each integer $r\geq 2$, there exists a constant $C_r>0$ with the following property: for any $0<\varepsilon \leq 1/2$ and any graph $G$ with clique number at most $r,$ there is a partition of $V(G)$ into at most $(1/\varepsilon)^{C_r}$ sets $S_1, \dots, S_t,$ such that $G[S_i]$ has maximum degree at most $\varepsilon |S_i|$ for each $1 \leq i \leq t.$ This answers a question of Fox, Nguyen, Scott and Seymour, who proved a similar result for graphs with no induced $P_4.$