The following concept arose when studying some properties of a real world computer network.

Let $G=(V,E)$ be a finite, simple, undirected graph. If $v\in V$ we set $N(v)=\{y\in V:\{v,y\}\in E\}$. Let $Z\neq \emptyset$ be a set. A map $c:V\to Z$ is said to be an "almost coloring" of $G$ if for all $v\in V$ the color of $v$ appears at most once in $N(v)$, that is, mathematically speaking, if $$|c^{-1}(\{c(v)\})\cap N(v)| \leq 1.$$ Let $\chi_a(G)$ denote the smallest integer $n$ such that there is an almost coloring $c:V\to\{1,\ldots,n\}$.

**Question**. If $G,H$ are finite, simple, undirected graphs with $\chi(G)\le \chi(H)$, does this imply that $\chi_a(G)\le\chi_a(H)$?

Also, I would be very grateful for a reference in case this concept that I call almost coloring has been studied elsewhere.