Past exam of the computer science course of the University of Oxford / 2021 / Artificial intelligence / 2 / e / Solution

This is a graph colouring constraint satisfaction problem, with four available colours and one fixed colour. Interpret slot labels as colours. Every conflict edge requires distinct endpoint colours, exactly as in proper graph colouring; independent sets of vertices can share a slot. The unary restriction on $A$ fixes its colour to $1$.

The lecture's 3-colouring problem is the same construction with domains $\{1,2,3\}$. For the present graph under the assumptions in (a), the clique $\{A,B,E,G\}$ prevents a 3-colouring, while (c) supplies a 4-colouring. The fixed colour on $A$ does not otherwise change existence, since an unconstrained colouring can have its colour names permuted to put $A$ in colour $1$.

This four-colour necessity is conditional on the explicit $EA$ overlap. Removing that one edge permits, for example, the 3-colouring $(A,B,C,D,E,F,G)=(1,2,2,3,1,2,3)$; the four-slot schedule remains valid in either model. Thus attendee modelling, rather than the mere availability of four slots, determines the required number of colours.