论文标题
在局部抗象征色素词素图图上
On local antimagic chromatic number of lexicographic product graphs
论文作者
论文摘要
令$ g =（v，e）$为连接的简单订单$ p $和size $ q $的图形。如果$ g $承认当地的抗魔法标签，则图$ g $称为本地抗原。 biftion $ f：e \ to \ {1,2，\ ldots，q \} $称为$ g $的本地抗害怕标签，如果对任何两个相邻的顶点$ u $和$ v $，我们都有$ f^+（u）\ ne f^+（u）\ ne f^+（v） $ e（u）$是事件的一组$ u $。因此，如果为顶点$ v $分配了颜色$ f^+（v）$，则任何本地抗原标签都会引起$ g $的适当顶点着色。本地的抗原色编号，表示为$χ_{la}（g）$，是$ g $的本地抗原标签所取的最小诱导颜色数量。令$ g $和$ h $为两个顶点脱节图。 {\ IT词典摄影产品}为$ g $和$ h $，表示为$ g [h] $，是带有顶点套装$ v（g）\ times v（h）$的图形，$（u，u'）$与$（v，v，v，v'）$相邻e（h）$。在本文中，我们获得了$χ_{la}（g [o_n]）$的尖锐上限，其中$ o_n $是$ n \ ge 1 $的空图。还获得了$χ_{la}（g）= 3 $的常规双分和三方图$ g $的足够条件。因此，我们成功地确定了局部的无限多（连接和断开连接的）常规图的本地反象征色素数，这些图是部分支持$ r $ ru $ ro $ r $ gragr $ g $的订单$ p $的$ p $，因此（i）$χ_{la}（g）=χ（g）=χ（g）= k $，以及（II）$ 1 = K $ 1 = K $ = ub ub ub ub $ r，p，k $。
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number of induced colors taken over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint graphs. The {\it lexicographic product} of $G$ and $H$, denoted $G[H]$, is the graph with vertex set $V(G) \times V(H)$, and $(u,u')$ is adjacent to $(v,v')$ in $G[H]$ if $(u,v)\in E(G)$ or if $u=v$ and $u'v'\in E(H)$. In this paper, we obtained sharp upper bound of $χ_{la}(G[O_n])$ where $O_n$ is a null graph of order $n\ge 1$. Sufficient conditions for even regular bipartite and tripartite graphs $G$ to have $χ_{la}(G)=3$ are also obtained. Consequently, we successfully determined the local antimagic chromatic number of infinitely many (connected and disconnected) regular graphs that partially support the existence of $r$-regular graph $G$ of order $p$ such that (i) $χ_{la}(G)=χ(G)=k$, and (ii) $χ_{la}(G)=χ(G)+1=k$ for each possible $r,p,k$.
