I am interested in a particular variant of (relaxed) list coloring. Let me call it neighbor-friendly coloring.
Let $G$ be a graph and $L:V(G)\to 2^\mathbb{N}$ be a function (of allowed colours).
An $L$-list coloring of $G$ is a coloring $f$ of $G$ such that $f(v)\in L(v)$ for each vertex $v$ of $G$ (that is, only colors in the set $L(v)$ can be used on $v$). In other words, a function $f:V(G)\to \mathbb{N}$ is an $L$-list coloring of $G$ if (i) $f(u)\neq f(v)$ for every edge $uv$ of $G$, and (ii) $f(v)\in L(v)$ for every vertex $v$ of $G$.
A function $f:V(G)\to \mathbb{N}$ is a neighbor-friendly $L$-coloring of $G$ if (i) $f(v)\in L(v)$ for every vertex $v$, and (ii) whenever $uv$ is an edge, colours used by $f$ on $u$ and $v$ are both in $L(u)$ OR both in L(v); that is, $f(u),f(v)\in L(u)$ OR $f(u),f(v)\in L(v)$. (Note that here we allow using same color at $u$ and $v$). The complexity of deciding whether $G$ has a neighbor-friendly $L$-coloring is baffling me even when $G$ is the complete graph $K_n$.
(I have tried attacking the problem along different lines such as expressing as SAT formula [a mixed horn formula], multi-colored clique problem and so on)
Is this variant already known? If not, have you come across something very similar or a problem which is obviously related to this?
(I remember seeing something very similar [but not relaxed coloring] in the literature; but couldn't trace it back at all. I have checked the chapter "List Colorings of Graphs" by Konrad Piwakowski in Kubale's book Graph Colorings, but found nothing similar)
Thank you