论文标题
在本地抗原顶点着色上
On local antimagic vertex coloring for complete full $t$-ary trees
论文作者
论文摘要
令$ g =（v，e）$为有限的简单无向图，而无需$ k_2 $组件。双重$ F： f（e）$和$ e（u）$是事件的一组$ u $。因此，任何局部的抗刺激标签都会引起$ g $的适当顶点着色，其中为顶点$ v $分配了颜色（vertex sum）$ w（v）$。本地抗原色数$χ_{la}（g）$是$ g $的本地抗原标签所引起的所有颜色所取的最小颜色数量。它是被猜想的\ cite {aru-wang}，对于每个树$ t $，本地抗原色编号$ l+ 1 \ leq uqχ_{la}（t）\ leq l+ 2 $，其中$ l $是$ t $的叶子的数量。在本文中，我们验证了上述猜想的完整$ t $ - 亚洲树，以$ t \ geq 2 $。完整的$ t $ - 亚洲树是一棵根树，在其中，所有节点都有$ t $儿童除叶子外，每片叶子都具有相同的深度。特别是，我们获得了所有完整的完整$ t $ are树的本地抗原色数的确切值为$ l+1 $，奇数$ t $。
Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e., $w(u) \neq w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color (vertex sum) $w(v)$. The local antimagic chromatic number $χ_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. It was conjectured \cite{Aru-Wang} that for every tree $T$ the local antimagic chromatic number $l+ 1 \leq χ_{la} ( T )\leq l+2$, where $l$ is the number of leaves of $T$. In this article we verify the above conjecture for complete full $t$-ary trees, for $t \geq 2$. A complete full $t$-ary tree is a rooted tree in which all nodes have exactly $t$ children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full $t$-ary trees is $ l+1$ for odd $t$.
