发表机构
The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文运用代数分析，得出了一类除3个顶点度数≥3外其余顶点度数均为2的二部图的局部反魔法色数为2的充要条件，并证明了对应诱导路径的连续边标号互不相同。
AI 中文摘要
对于连通图$G=(V,E)$，若双射边标号$f:E\to\{1,\ldots,|E|\}$是$G$的局部反魔法标号，则其诱导顶点标号$f^+$满足：对任意相邻顶点$x$和$y$，$f^+(x)\neq f^+(y)$，其中诱导顶点标号$f^+(x)$为$x$的所有相邻顶点$u$对应的边标号$f(xu)$之和。$G$的所有局部反魔法标号中不同诱导顶点标号的最小数量称为局部反魔法色数，记为$\\_\text{la}(G)$。本文利用代数分析，得到了一类二部图的充要条件：该类二部图除恰好3个顶点的度数至少为3外，其余所有顶点的度数均为2，且这类二部图的局部反魔法色数为2。此外，本文还证明了每种情形下各诱导路径的连续边标号均互不相同。
英文摘要
For a connected graph $G = (V, E)$, a bijective edge labeling $f:E \to\{1,\ldots ,|E|\}$ is a local antimagic labeling of $G$ if it induces a vertex labeling $f^+$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(xu)$, with $u$ ranging over all the vertices adjacent to $x$. The minimum number of distinct induced vertex labels over all local antimagic labelings of $G$ is the local antimagic chromatic number of $G$, denoted $χ_{la}(G)$. In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least $3$ (say, $u,v,w$, such that every $(u,v)$-, $(u,w)$- and $(v,w)$-path is of even length and not all are of length 2) to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.
CommentsIn this version, we revised the abstract to include the correct conditions imposed on our graphs
