AI 中文总结
本文针对局部全邻域反魔法标号的色数问题，提出新方法并确定了阶数奇偶性不同的图的联图的该色数，为图论标号相关研究提供了新结果。
AI 中文摘要
给定一个阶为$p$、大小为$q$的简单图$G=(V,E)$，若双射$f: V\bigcup E \to \{1, 2, \ldots, p+q\}$满足：对任意相邻的两个顶点$u$和$v$，其诱导顶点着色具有性质$f^+_{tn}(u) \ne f^+_{tn}(v)$，其中$f^+_{tn}(u)$是$u$的所有邻点$x$对应的$(f(ux) + f(x))$之和，则称该双射为图$G$的局部全邻域反魔法标号。图$G$的局部全邻域反魔法色数记为$\\_\chi_{ltna}(G)$，其定义为在图$G$的所有局部全邻域反魔法标号中，诱导颜色的不同数量的最小值。本文中，我们确定了阶数奇偶性不同的图的联图的局部全邻域反魔法色数。
英文摘要
Given a simple graph $G=(V,E)$ of order $p$ and size $q$, a bijection $f : V\cup E \to \{1, 2, \ldots, p+q\}$ is a local total neighborhood antimagic labeling of $G$ if the induced vertex coloring has the property $f^+_{tn}(u) \ne f^+_{tn}(v)$ for every two adjacent vertices $u$ and $v$ where $f^+_{tn}(u) = \sum (f(ux) + f(x))$ over every neighbor $x$ of $u$. The local total neighborhood antimagic chromatic number of $G$, denoted $χ_{ltna}(G)$ is the minimum number of distinct induced colors over all local total neighborhood antimagic labeling of $G$. In this paper, we determine the local total neighborhood antimagic chromatic number of the join of graphs with distinct parity orders.
Commentssubmitted to Acta Mathematica Hungarica
