Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 38 / 4 / c / Solution

The displayed graph consists of three graph triangles sharing the vertex $v$. In a graph colouring with three colours, every graph triangle uses all three colours. The edge $tf$ and the edges $tv,fv$ therefore force $c(t),c(f)$ to be the two colours different from $c(v)$. Likewise the graph triangle $vxy$ forces $c(x),c(y)$ to be those same two colours. The third graph triangle imposes no further restriction on these four vertices. HenceBoth two-element sets and their union have cardinality two. This is a Boolean-pair colouring gadget: after fixing a palette graph triangle, the pair $x,y$ encodes opposite truth values.