Definition of $\mathbb{Z}_2$-periodic graph I see a definition of a planar, bipartite $\mathbb{Z}_2$-periodic graph, which is a graph can be embedded in the plane, the vertices can be coloured by white and black such that every edges of the graph has exactly one white vertex and one black vertex, and all the translations act by colour-preserving isomorphisms, i.e. the isomorphisms which map white vertices to white ones and black vertices to black ones.
I don't understand the last condition. Someone can explain clearly?
Thanks a lot!
 A: It simply means that, in the case $\mathbb{Z}_2$-periodic graphs, we only consider those graph-isomorphisms which preserve vertex colour to be a $\mathbb{Z}_2$-periodic-graph-isomorphism. For instance, consider the graph $G$ which looks like $\bullet-\circ$ With black vertex $b$ and white vertex $w$.
The map $f\colon G\rightarrow G$ which maps $b\mapsto w$,$w\mapsto b$ is a graph isomorphism, but it is not a $\mathbb{Z}_2$-periodic-graph-isomorphism because it does not preserve vertex colour. In the case of $G$, the only $\mathbb{Z}_2$-periodic-graph-isomorphism is the identity map.
Let $G'$ be the graph which looks like $\bullet-\circ -\bullet$ With black vertices $b_1, b_2$ and white vertex $w$. In this case, the map $f'\colon G'\rightarrow G'$ which maps $b_1\mapsto b_2$, $b_2\mapsto b_1$, $w\mapsto w$ is both a graph isomorphism and a $\mathbb{Z}_2$-periodic-graph-isomorphism because it does preserve vertex colour.
For a more complex example, Consider the integer lattice $\mathbb{Z}^2$ generated by the vectors $(0,1)$ and $(1,0)$ and colour the vertex $(a,b)$ black if $a+b$ is even, and white if $a+b$ is odd - this is the classic checkerboard colouring of $\mathbb{Z}^2$. Add an edge between $(a,b)$ and $(a',b')$ if $a+1=a'$ or $b+1=b'$ and not both $a+1=a'$ and $b+1=b'$. We note that  this defines a planar bipartite $\mathbb{Z}_2$-periodic graph $G$. Consider the map $t\colon G''\rightarrow G''$ defined by $t(a,b)=(a+1,b)$. This is a graph isomorphism but is clearly not colour preserving. On the other hand, the map $t'\colon G''\rightarrow G''$ defined by $t'(a,b)=(a+1,b+1)$ is colour preserving and so it is a $\mathbb{Z}_2$-periodic-graph-isomorphism.
