The additive 1-2 Conjecture

Let $G$ be a graph, and let $\chi(G)$ denote its chromatic number. For a vertex decoration $f$, define $S(v)=\sum_{x\in N(v)}f(x)$, and call $f$ cool if $S(u)\ne S(v)$ for every adjacent pair $u,v$. The additive 1-2 Conjecture. Every graph $G$ has a cool decoration of vertices from the set $\{1,2,\dots,\chi(G)\}$. The assertion is tight for cliques and remains open even for bipartite graphs.