Unresolved 3-Colouring cases on P_t-free graphs

Determine the computational complexity of 3-Colouring on P_t-free graphs for every t≥8.

Background

The paper summarizes known polynomial-time results for 3-Colouring on P_t-free graphs up to t=7 and notes a quasi-polynomial-time algorithm for larger t. Despite this progress, the complexity classification for t≥8 has not been established.

This unresolved H-free problem provides context for the paper’s study of probe P_t-free graphs, where the threshold for partitioned probe graphs is shown to be t=5 versus t=6.

References

The cases $k=3$ and $t\geq 8$ are still open, despite some evidence that these cases are polynomial-time solvable due to a quasi-polynomial-time algorithm.

— Colouring Probe $H$-Free Graphs  (2505.20784 - Paulusma et al., 27 May 2025) in Section 1, Introduction, subsection “H-Free Graphs”