Weak coloring number characterization of radius-2 merge-width

Determine whether the radius-2 merge-width of a graph is bounded in terms of its weak 2-coloring number.

Background

The paper proves that radius-fr merge-width is bounded in terms of the weak f(r+1)-coloring number. For radius 2, this gives a bound using the weak 3-coloring number, but the authors leave open whether the weaker weak 2-coloring number suffices.

References

In particular, we show that \mw_2(G) is bounded in terms of \wcol_3(G), and we do not know whether it is bounded in terms of \wcol_2(G).

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section 2, Lemma \(\ref{lem:wcol}\) and its concluding paragraph