k-Colouring on partitioned probe P_5-free graphs for k≥4

Determine the computational complexity of k-Colouring on partitioned probe P_5-free graphs for every fixed integer k≥4.

Background

The paper extends the known polynomial-time solvability of k-Colouring on P_5-free graphs to 3-Colouring on partitioned probe P_5-free graphs. It explicitly leaves open whether this extension holds for all fixed k≥4, noting that structural properties used in the 3-colouring proof no longer necessarily apply.

References

For $k \geq 4$, what is the complexity of {$k$} on partitioned probe $P_5$-free graphs?

— Colouring Probe $H$-Free Graphs  (2505.20784 - Paulusma et al., 27 May 2025) in Section 6, “Additional Results and Concluding Remarks,” immediately after the open cases for 3-Colouring

Since {$k$} is polynomial on $P_5$-free graphs even for all $k\geq 3$, we ask:

For $k \geq 4$, what is the complexity of {$k$} on partitioned probe $P_5$-free graphs?

— Colouring Probe $H$-Free Graphs  (2505.20784 - Paulusma et al., 27 May 2025) in Section 6, Additional Results and Concluding Remarks