Coloring complexity on monoholed graphs

Determine the computational complexity of Coloring on monoholed graphs.

Background

The paper establishes that monoholed graphs have bounded strong isometric path complexity and notes that Maximum Independent Set and certain clique problems are polynomial-time solvable on related classes. In contrast, the complexity of Coloring is explicitly left unresolved for the full class of monoholed graphs.

References

The computational complexity of Coloring is open on these graphs.

— Strong isometric path complexity of graphs: Asymptotic minors, restricted holes, and graph operations  (2501.10828 - Chakraborty et al., 18 Jan 2025) in Section 1, subsection “Monoholed graphs and even-hole free graphs”