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

We investigate the relationship between graph parameters, which measure the complexity of the tree decompositions of a given graph. The treewidth 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 tree-α(G) and the tree-chromatic number tree-χ(G) measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of G. Recently, Dallard, Milanič, and Štorgel asked (JCTB, 2024) whether for all graphs G it holds that tw(G)+1 ≤ tree-α(G) ⋅ tree-χ(G). We provide a negative answer for this question in a strong form: for every function f: {ℕ} → {ℕ}, there exists a graph G such that tw(G) > tree-α(G) ⋅ f(tree-χ(G)). On the other hand, we complement this result with an upper bound, by showing that tw(G)+1 ≤ tree-α(G)² ⋅ tree-χ(G) for every graph G.