关于可加集值符号图的一些新结果
Some New Results on Integer Additive Set-Valued Signed Graphs
摘要：
设$X$表示一个非负整数集，$\mathscr{P}(X)$表示它的幂集.图$G$的整数可加集标号（IASL）是一个内射集值函数$f:V(G)\to \mathscr{P}(X)-\{\emptyset\}$，使得导出函数$f^+:E(G) \to \mathscr{P}(X)-\{\emptyset\}$由$f^+(uv)=f(u)+f(v);\ \forall\, uv\in E(G)$定义，其中$f(u)+f(v)$是$f(u)$和$f(v)$的和集.有符号图的IASL是其底层图$G$连同由$σ(uv)=(-1)^{|f^+(uv)|};\ \forall\, uv\in E(Σ)$定义的签名$σ$的IASL。本文讨论了允许某些类型的整数可加集合标号的符号图的某些特征。
英文摘要：
Let $X$ denotes a set of non-negative integers and $\mathscr{P}(X)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \mathscr{P}(X)-\{\emptyset\}$ is defined by $f^+(uv)=f(u)+f(v);\ \forall\, uv\in E(G)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. An IASL of a signed graph is an IASL of its underlying graph $G$ together with the signature $σ$ defined by $σ(uv)=(-1)^{|f^+(uv)|};\ \forall\, uv\in E(Σ)$. In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings.
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