Colour refinement algorithm

We define a sequence of vertex colourings ${\displaystyle cr^{t}}$ defined as follows:

• ${\displaystyle cr^{0}}$is the initial colouring. If the graph is unlabeled, the initial colouring assigns a trivial color ${\displaystyle cr^{0}(v)}$ to each vertex ${\displaystyle v}$. If the graph is labeled, ${\displaystyle cr^{0}(v)}$ is the label of vertex ${\displaystyle v}$.
• For all vertices ${\displaystyle v}$, we set ${\displaystyle cr^{t+1}(v)=\left(cr^{t}(v),\{\{cr^{t}(w)\mid w{\text{ is a neighbor of }}v\}\}\right)}$. In other words, the new color of the vertex ${\displaystyle v}$ is the pair formed from the previous color and the multiset of the colors of its neighbors.

This algorithm keeps refining the current colouring. At some point, before n steps, where n is the number of vertices, it stabilises: ${\displaystyle cr^{t}}$does not change anymore when t increases. This final colouring is called the stable colouring.

This algorithm does not distinguish a cycle of length 6 from a pair of triangles (example V.1 in [2]).

The stable colouring is computable in O((n+m)log n) where n is the number of vertices and m the number of edges.[3] This complexity has been proven to be optimal for some class of graphs.[4]