论文标题
关于局部抗怪异色素词素产品图的注释
A note on local antimagic chromatic number of lexicographic product graphs
论文作者
论文摘要
令$ g =（v，e）$为连接的简单图。 Biftion $ f：e \ rightarrow \ {1,2，\ ldots，| e | \} $称为$ g $的本地反象征标签，如果$ f^+（u）\ neq f^+（v+（v）$都持有任何两个相邻的$ $ u $和$ v $ u $和$ v $，wher $ e（u $）是事件的一组边缘。如果$ g $至少承认当地的反刺激标签，则图$ g $称为本地抗原。本地的抗原色编号，表示为$χ_{la}（g）$，是$ g $的本地抗原标签所取的诱导颜色的最低数量。令$ g $，$ h $为两个不相交的图。图$ g [h] $是由$ g $和$ h $的词典产物获得的。在本文中，我们获得了$χ_{la}（g [h]）\ leqχ_{la}（g）χ_{la}（h）$的足够条件。因此，我们提供了$ g $和$ h $的示例，以便$χ_{la}（g [h]）=χ（g）χ（h）$，其中$χ（g）$是$ g $的色数。我们认为（i）有很多图形$ g $和$ h $，因此$χ_{la}（g [h]）=χ_{la}（g）χ_{la}（la}（h）=χ（g）χ（g）χ（h）$和（ii）的$ k \ ge 1 $，$，$，$，$ c _ $ c = h（H）只有$χ（g）χ（h）=2χ（h） + \ lceil \ frac {χ（h）} {k} {k} \ rceil $，其中$ 2K + 1 $是$ g $中最短奇数周期的长度。
Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained by the lexicographic product of $G$ and $H$. In this paper, we obtain sufficient conditions for $χ_{la}(G[H])\leq χ_{la}(G)χ_{la}(H)$. Consequently, we give examples of $G$ and $H$ such that $χ_{la}(G[H]) = χ(G)χ(H)$, where $χ(G)$ is the chromatic number of $G$. We conjecture that (i) there are infinitely many graphs $G$ and $H$ such that $χ_{la}(G[H])=χ_{la}(G)χ_{la}(H) = χ(G)χ(H)$, and (ii) for $k\ge 1$, $χ_{la}(G[H]) = χ(G)χ(H)$ if and only if $χ(G)χ(H) = 2χ(H) + \lceil\frac{χ(H)}{k}\rceil$, where $2k+1$ is the length of a shortest odd cycle in $G$.
