Extremal List Gaps and Inapproximability in Additive Graph Labeling
We study a vertex-labeling analogue of the $1$-$2$-$3$ problem and its list version. For a labeling $\ell:V(G)\to\mathbb N$, let $S_\ell(v)=\sum_{w\in N(v)}\ell(w)$. The additive number $η(G)$ is the least $k$ for which there exists $\ell:V(G)\to[k]$ such that $S_\ell(u)\ne S_\ell(v)$ for every $uv\in E(G)$, while the list additive number $η_\ell(G)$ is the least $k$ such that the same condition can be satisfied from every assignment of $k$-element lists $L(v)\subset\mathbb N$ with $\ell(v)\in L(v)$. We show that for every $k\ge2$, there is a graph $G$ with $η(G)=1$ and $η_\ell(G)\ge k$. The separation persists at the minimum possible ordinary value for positive-degree regular graphs: there is a regular graph $H$ with $η(H)=2$ and $η_\ell(H)\ge k$. We also determine a sharp lower bound for $η(G)$ in terms of the order and minimum degree of $G$, and show that the unbounded list gap persists at asymptotically extremal density. Finally, for every fixed $k\ge2$, it is NP-hard to distinguish $η(G)=2$ from $η(G)>k$, even on asymptotically extremal dense graphs. Consequently, $η(G)$ admits no polynomial-time constant-factor approximation unless $\mathrm P=\mathrm{NP}$. Together, these results reveal a robust gap phenomenon: the separation between ordinary and list additive labeling persists at the smallest possible ordinary values and even under asymptotically extremal density, while the ordinary parameter itself remains hard to approximate.