= 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. Hence
$$
\boxed{\{c(t),c(f)\}=\{c(x),c(y)\},\qquad c(t)\ne c(f),\quad c(x)\ne c(y).}
$$
Both 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.
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