I have an infinite graph $G^\infty$ constructed out of sequence $G_t$ of copies of some finite graph $G$. More specifically:

- Vertex set of $G^\infty$ is $$V(G^\infty) = \bigcup_{i \in \mathbb{Z}} V(G_i)$$
- Edge set of $G^\infty$ is $$E(G^\infty) = \bigcup_{i \in \mathbb{Z}} \left( E(G_i) \cup \bigcup_{j = 1}^K \hat{E}_j(G_i, G_{i+j}) \right)$$ Where $\hat{E}_j(G_i, G_{i+j})$ is arbitrary set of edges between vertices of $G_i$ and $G_{i+j}$ and $K$ is some finite constant. Note that $\hat{E}_j$ doesn't depend on $i$, so $G^\infty$ has a regular structure.

I want to find a good proper vertex coloring of such graph. So far I found the following simple algorithm:

- Join $G_0, \ldots, G_{K}$ graphs circularly by replacing $\hat{E}_j(G_i, G_{i+j})$ with $\hat{E}_j(G_i, G_{(i+j)\ \mathrm{mod}\ (K+1)})$
- Color joined graph with greedy coloring
- To color $G_i$ from $G^\infty$ use coloring of $G_{i\ \mathrm{mod}\ (K+1)}$ from joined graph

But I think a better coloring algorithm can be found. So the question is there any known coloring algorithms for these types of graphs? And do these types of graphs has a common name and mentioned in literature?