Coloring graph with 3 colors The nodes of the graph $[V, E]$ are the lists $\left[ {{s_1},\,{s_2},\,{s_3}} \right]$ where ${s_j} \in \left\{ {1,2} \right\},\,\,j = 1,2,3$.
There is an undirected edge between the nodes $\left[ {{s_1},\,{s_2},\,{s_3}} \right]$ and $\left[ {{t_1},\,{t_2},\,{t_3}} \right]$ if and only if $|{s_1} - {t_1}| + |{s_2} - {t_2}| + |{s_3} - {t_3}|\,\, = \,\,1$. Draw this graph and put the nodes in an order so that the greedy coloring algorithm uses $3$ colors.
My attempt:
I got the following graph:

Now i don't know how to put the nodes in an order so that the greedy coloring algorithm will use $3$ colors instead of obvious $2$ colors.

Lets order the vertices $(211), (111), (122),(112),(212), (222), (221), (121)$
Using the colours $1,2,3$, the greedy algorithm assigns the colours $1$ to $(221)$, $2$ to $(111)$, $1$ to $(122)$, $3$ to $(112)$, $2$ to $(212)$, $3$ to $(222)$, $2$ to $(221)$, and $3$ to $(121)$, giving you a $3$-colouring.
