加性图标记中的极值列表间隙与不可近似性
Extremal List Gaps and Inapproximability in Additive Graph Labeling
- Tehran Institute for Advanced Studies (TeIAS), Khatam University, Iran（德黑兰高等研究院（TeIAS），哈塔姆大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究加性图标记中普通加性数与列表加性数之间的分离，证明即使在最小普通值及渐近极值密度下间隙仍无界，且普通加性数无法多项式时间常数因子近似。
中文摘要 AI 辅助
我们研究了$1$-$2$-$3$问题的顶点标记类比及其列表版本。对于标记$\ell:V(G)\to\mathbb N$，令$S_\ell(v)=\sum_{w\in N(v)}\ell(w)$。加性数$\eta(G)$是使得存在$\ell:V(G)\to[k]$且对每条边$uv\in E(G)$有$S_\ell(u)\ne S_\ell(v)$的最小$k$，而列表加性数$\eta_\ell(G)$是使得从每个$k$元列表$L(v)\subset\mathbb N$的赋值（其中$\ell(v)\in L(v)$）都能满足相同条件的最小$k$。我们证明对于每个$k\ge2$，存在图$G$满足$\eta(G)=1$且$\eta_\ell(G)\ge k$。对于正度正则图，这种分离在最小可能的普通值处仍然存在：存在正则图$H$满足$\eta(H)=2$且$\eta_\ell(H)\ge k$。我们还确定了$\eta(G)$关于$G$的阶和最小度的锐利下界，并证明无界列表间隙在渐近极值密度下仍然存在。最后，对于每个固定的$k\ge2$，即使在渐近极值稠密图上，区分$\eta(G)=2$与$\eta(G)>k$也是NP困难的。因此，除非$\mathrm P=\mathrm{NP}$，$\eta(G)$不存在多项式时间常数因子近似。这些结果共同揭示了一个稳健的间隙现象：普通与列表加性标记之间的分离在最小可能的普通值处以及渐近极值密度下仍然存在，而普通参数本身仍难以近似。
英文摘要
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.
