Z/2Z and Bipartite graph

Bipartite graph has an interesting connection to ${\mathbb{F}}_2(\mathbb{Z}/2\mathbb{Z})$$\mathbb F_2(\mathbb Z/2\mathbb Z)$

We know that a connected graph is bipartite if and only if it does not have an odd-length loop.

And we can use ${\mathbb{F}}_2$$\mathbb F_2$ to prove it!

We can choose one vertex in a connected graph and give the vertex a value of 0.

Since it is connected, and we view the edges as $+1$$+1$, so another vertex connected with the initial vertex is 1.

If there exists an odd-length loop, then all the vertex value in this loop is not well-defined.

Because an odd number is $1$$1$ in ${\mathbb{F}}_2$$\mathbb F_2$, and $0+1\ne 0,1+1\ne 1$$0+1\ne 0,1+1\ne 1$ .

If there is no odd-length loop in this connected graph, then all the point is well-defined

The vertex has been divided into two equivalence classes $[0{]}_2,[1{]}_2$$[0]_2,[1]_2$, and the edges $+1$$+1$ connected from $[0{]}_2$$[0]_2$ to$[1{]}_2$$[1]_2$