论文标题
用局部小型抗原色编号无限输出许多图的方法
Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number
论文作者
论文摘要
连接图的边缘标记$ g =（v，e）$，如果是二维$ f：e \ to \ to \ {1，\ ldots，| e | \} $，则是局部反象征，因此对于任何一对相邻的顶点$ x $ x $ and $ x $ and $ x $，$ f^+（x） \ sum f（e）$，$ e $在所有事件的边缘范围内到$ x $。 $ g $的本地抗原色数，用$χ_{la}（g）$表示，是所有本地抗原标签上$ g $的最小诱导顶点标签的最小数量。在本文中，我们（i）给出了一个带有一个吊坠的图形的条件，使其具有$χ_{la} \ ge 3 $。然后获得图形具有$χ_{la} = 2 $的必要条件； （ii）为每个循环图提供了足够的条件，即使订单具有$χ_{la} = 3 $； （iii）通过循环的转换，构建了无限的许多两分和三方图，构建$χ_{la} = 3 $； （iv）应用循环的转换以获得无限的许多单点联合（可能是循环体）或双向图形，$χ_{la} = 2,3 $。本文的工作提出了关于局部抗原色的两部分和三方图的许多开放问题。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we (i) give a sufficient condition for a graph with one pendant to have $χ_{la}\ge 3$. A necessary and sufficient condition for a graph to have $χ_{la}=2$ is then obtained; (ii) give a sufficient condition for every circulant graph of even order to have $χ_{la} = 3$; (iii) construct infinitely many bipartite and tripartite graphs with $χ_{la} = 3$ by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with $χ_{la} = 2,3$. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.
