C2/C3 status of Sequential 3-Colouring Construction Game in the C123 framework

Determine whether Sequential 3-Colouring Construction Game satisfies conditions C2 and C3 of the finitely-bounded monotone-class framework; specifically, establish whether the game is computationally hard on subcubic graphs (C2) and whether this hardness is preserved under edge subdivision of subcubic graphs (C3).

Background

The authors show Pspace-completeness of Sequential 3-Colouring Construction Game on bounded pathwidth by relating it to QCSP(K3). However, they do not know whether it meets the framework’s hardness requirements C2 and C3.

Clarifying these conditions would determine whether this game admits the same kind of dichotomy results under the framework as established for several other problems.