Hi, everyone, I want to ask a question about graph theory.

Let $G$ be a finite graph, and $E$ the set of edges of $G$. For each vertex $a$, we denote that $E_{a}$ by the edges of $G$ which pass to $a$. Suppose for any $a \in G$, $\sharp E_{a} >3$. Let $f: E \longrightarrow$ {$1,2$}.

Does there exists a map $f$ such that for each $a$, $\sharp (f^{-1}(1) \cap E_{a})$ is a nonzero even number.

Thank you very much!