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C2/C3 status of QCSP(K3) in the C123 framework

Determine whether QCSP(K3) satisfies conditions C2 and C3 of the finitely-bounded monotone-class framework; specifically, establish whether QCSP(K3) is computationally hard on subcubic graphs (C2) and whether this hardness is preserved under edge subdivision of subcubic graphs (C3).

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Background

The paper proves that QCSP(K3) is Pspace-complete on bounded pathwidth, but it does not settle whether QCSP(K3) fulfills the defining hardness conditions C2 (hardness on subcubic graphs) and C3 (hardness preserved under edge subdivisions) required for C123-problems.

The authors explicitly leave this as open future work, indicating that resolving these conditions would clarify how QCSP(K3) fits into their framework of dichotomies on monotone classes.

References

Moreover, we proved hardness for bounded path-width for QCSP$(K_3)$ and {\sc Sequential $3$-Colouring Construction Game}. We do not know if the latter two problems satisfy C2 and C3. We leave this for future work.

— Graph Homomorphism, Monotone Classes and Bounded Pathwidth (2403.00497 - Eagling-Vose et al., 1 Mar 2024) in Conclusions