Graphs of bounded cliquewidth are polynomially $χ$-bounded

Abstract

<jats:p>We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $\chi$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $\chi$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $\chi$-bounded.</jats:p>

Rankwidth meets stability

“…Moreover, eorem 1.2 has important corollaries for classes with low rankwidth covers/colorings (introduced in [29]). It follows from [5] Our results together with observations present in the literature are illustrated by the semi-la ice of properties of graph classes in Figure 1. See Figure 5 in Section 6 for an extended version of the schema.…”

“…In this work we prove the conjecture stated in [34] e implications 4⇒3⇒2⇒1 are obvious. For the implication 1⇒4, we combine the approach presented in [34] with the techniques used by Bonamy and the third author in [5] to prove that classes of bounded rankwidth are polynomially χ-bounded. Using the tree variant of Simon's factorization due to Colcombet [7], the authors of [5] introduce a bounded-depth recursive decomposition of the tree encoding of a graph of rankwidth at most k into factors, so that the quotient trees satisfy certain Ramsey properties.…”

A survey of χ‐boundedness

“…For example, the closure of an ideal under the operation of gluing along at most vertices (where the two graphs being glued together have the same induced subgraph on the overlap) preserves ‐boundedness (this is proved in [25], or can be deduced from earlier work of Alon, Kleitman, Saks, Seymour, and Thomassen [8]). The closure of a ‐bounded ideal under 1‐joins is also ‐bounded (see Dvořák and Král' [45], Bonamy and Pilipczuk [17], and Kim, Kwon, Oum, and Sivaraman [77]). It would be interesting to know what happens with other graph compositions (see [25] for discussion).…”

“…Some ideals of graphs are known to be polynomially ‐bounded (see, eg, Schiermeyer and Randerath [104] and Bonamy and Pilipczuk [17]). But it is easy to see that there is no such that every ‐bounded ideal is bounded by a polynomial of degree .…”

The Erdős-Hajnal Property for Graphs with No Fixed Cycle as a Pivot-Minor

“…Theorem 30 (Bonamy and Pilipczuk [3]). For each k, the class of graphs of rank-width at most k is polynomially χ-bounded.…”