On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs

We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth ${\rm tw}(G)$ of a graph $G$ measures the largest number of vertices required in a bag of every tree decomposition of $G$. Similarly, the tree-independence number ${\rm tree\textnormal{-}}\alpha(G)$ and the tree-chromatic number ${\rm tree\textnormal{-}}\chi(G)$ measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of $G$. Recently, Dallard, Milani\v{c}, and \v{S}torgel asked (JCTB, 2024) whether for all graphs $G$ it holds that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G) \cdot {\rm tree\textnormal{-}}\chi(G)$. We provide a negative answer for this question in a strong form: for every function $f\colon {\mathbb N} \rightarrow {\mathbb N}$, there exists a graph $G$ such that ${\rm tw}(G) > {\rm tree\textnormal{-}}\alpha(G) \cdot f({\rm tree\textnormal{-}}\chi(G))$. On the other hand, we complement this result with an upper bound, by showing that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G)^2 \cdot {\rm tree\textnormal{-}}\chi(G)$ for every graph $G$.