论文标题
在局部抗刺激色的蜘蛛图上
On Local Antimagic Chromatic Number of Spider Graphs
论文作者
论文摘要
连接图的边缘标记$ g =（v，e）$，如果是二线$ f：e \ to \ {1，。 。 。 ，| e | \} $，使得对于任何一对相邻的顶点$ x $和$ y $，$ f^+（x）\ ne f^+（y）$，其中诱导的dertex标签$ f^+（x）= \ sum f（e）$，$ e $在所有edges事件上均以$ e范围到$ 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，我们首先证明A $ D $ -LEG SPIDER图具有$ D+1 \leχ_{la} \ le D+2 $。然后，我们获得了许多足够的条件，以使两个值都是可以实现的。最后，我们证明每个3腿蜘蛛都有$χ_{la} = 4 $，如果不是所有的腿都奇怪。我们推测，几乎所有满足$ d（d+1）\ le 2（2q-1）$的$ q $的$ D $ -Leg蜘蛛，每个腿长至少2个具有$χ_{la} = D+1 $。
An edge labeling of a connected graph $G = (V,E)$ is said to be local antimagic if it is a bijection $f : E \to \{1, . . . , |E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \ne f^+(y)$, 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 first show that a $d$-leg spider graph has $d+1\le χ_{la}\le d+2$. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has $χ_{la} = 4$ if not all legs are of odd length. We conjecture that almost all $d$-leg spiders of size $q$ that satisfies $d(d+1) \le 2(2q-1)$ with each leg length at least 2 has $χ_{la} = d+1$.
