Dynamic chromatic number versus treewidth and dynamicity parameter

Determine whether the r-dynamic chromatic number satisfies χ_r(G) = O(tw(G) + r) for every graph G, improving the established upper bound χ_r(G) ≤ tw(G)·r + 1.

Background

The paper proves that every graph G satisfies χ_r(G) ≤ tw(G)·r + 1 by bounding the strong 2-coloring number in terms of treewidth and then applying the main dynamic-coloring theorem. The authors explicitly note that they know no example attaining this multiplicative upper bound.

The unresolved question asks whether a substantially stronger additive dependence on treewidth and r holds universally: namely, whether the number of colors needed for an r-dynamic coloring can always be bounded by a constant multiple of tw(G) + r, rather than by a product tw(G)·r.