图的最小和标号
Minimal Sum Labeling of Graphs
摘要：
图$G$称为和图，如果存在$G$的所谓和标记，即内射函数$\ell: V(G) \rightarrow \mathbb{N}$使得对于每个$u,v\in V(G)$，它保持$uv\in E(G)$当且仅当存在顶点$w\in V(G)$使得$\ell(u)+\ell(v) = \ell(w)$。如果存在一个顶点$\ell$使得$\ell(u)=1$ .在本文中，我们表明，如果我们放松的条件（无论是允许非内射标号或考虑图与循环），那么有和图没有一个最小的标签，这部分回答了问题所提出的米勒，瑞安和史密斯在1998年。
英文摘要：
A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists a vertex $w\in V(G)$ such that $\ell(u)+\ell(v) = \ell(w)$. We say that sum labeling $\ell$ is minimal if there is a vertex $u\in V(G)$ such that $\ell(u)=1$. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth in 1998.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
