I am trying to read An Optimal Lower Bound on the Number of Variables for Graph Identification. On page 3 (4th paragraph), it is written-
It might color vertices and edges implicitly by using $k$-tuples with repetition of components. It could start with some encoding of the graph into the labels assigned to the $k$-tuples. For example, the initial label or color of every $k$-tuple could be the number of its distinct components except when this number is two. Then two colors could be used to encode the presence or absence of an edge between the two vertices. We prefer to get a quicker start by initially coloring each $k$-tuple with its isomorphism type. Repeatedly, the color of $(u_1, . . . , u_k)$ is refined by the $n$ element multiset (containing one element for each vertex $v$) of $k$-tuples of colors previously assigned to
$(v, u_2, . . . , u_k), (u_1, v, u_3, . . . , u_k), . . . , (u_1, . . . , u_{k−1}, v)$
We only need to consider $k$-tuples of distinct elements, if we finally color the vertices by the multiset of colors of incident $k$-tuples.
I am having hard time get this passage. I would appreciate some help about the coloring with some example. Note that, if $k=1$ , then it is simple vertex classification (partition based on degree).