number of induced subgraph for a prime number $p$

Question

For a graph $G$ and a prime number $p$, show that the number of induced subgraphs, that $p|deg(v)$ for each vertex of these subgraphs, is divisible by $p$.

I wanted to use Chevalley-Warning theorem, but it doesn't work.

Answer 1

I think this is false such that Misha Lavrov said.

But I think this is true for just $p=2$. Because suppose $A$ is the incidence matrix of graph $G$ in $\mathbb{z}_2$. any vector in $ker(A)$ is a $\{0,1\}$ vector. That means we choose some edges which this number is even. We know $dim(ker(A))=m-n+c$, where $m, n, c$ is the number of edges, vertices and connected component, respectively. This is a famous result that there exists $m-n+c$ cycles in $G$ such that their corresponding vectors form a basis for $ker(A)$. So it is obvious that the union of some of these cycles in $ker(A)$ and conversely. The number of it is $2^{m-n+c}$. This is even.