论文标题
在局部抗原色数中的吊坠数量
On number of pendants in 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 $的最小诱导顶点标签的最小数量。令$χ（g）$为$ g $的色数。在本文中，获得$ g $的$χ_{la}（g）$的尖锐上限和下限，并获得了足够的条件，并获得了足够的条件使得相等。因此，对于$ k \ ge 1 $，有许多图形，带有$ k \geχ（g）-1 $吊坠顶点，$χ_{la}（g）= k+1 $。我们推测，除某些毛毛虫，蜘蛛和龙虾外，每棵树$ t_k $都有$ k \ ge 1 $ pendant顶点具有$χ_{la}（t_k）= k+1 $。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ 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 $G$. Let $χ(G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of $χ_{la}(G)$ for $G$ with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for $k\ge 1$, there are infinitely many graphs with $k \ge χ(G) - 1$ pendant vertices and $χ_{la}(G) = k+1$. We conjecture that every tree $T_k$, other than certain caterpillars, spiders and lobsters, with $k\ge 1$ pendant vertices has $χ_{la}(T_k) = k+1$.
