Characterization of clustered coloring number in minor-closed classes

Let $t\geq 1$ be an integer. For a graph class $\mathcal{G}$, let $\operatorname{col}_\star(\mathcal{G})$ be the least integer $t$ for which there is a constant $C\geq 1$ such that every graph $G\in\mathcal{G}$ has $\operatorname{col}_C(G)\leq t$. The graphs $K_{t,m}$ are complete bipartite graphs, and $I_{t-1}+P_m$ is the graph formed from a path on $m$ vertices by adding $t-1$ vertices adjacent to every path vertex. Conjecture on clustered coloring number. A minor-closed class of graphs $\mathcal{G}$ satisfies

$$\operatorname{col}_\star(\mathcal{G})\leq t$$

if and only if there exists $m\geq 1$ such that

$$K_{t,m}\notin\mathcal{G}\quad\text{and}\quad I_{t-1}+P_m\notin\mathcal{G}.$$

The preceding observation shows that the condition is necessary: bounded clustered coloring number excludes all but finitely many graphs of these two forms. The conjecture asserts sufficiency for minor-closed classes, giving an exact characterization; the paper presents it as an open question.