关于不交并星的公平选择性
On the Equitable Choosability of the Disjoint Union of Stars
摘要：
公平$k$ -可选性是公平$k$ -着色的列表类似物，由Kostochka，Pelsmajer和West在2003年引入。已知如果顶点不交图$G_1$和$G_2$是$k$可公平选择的，则$G_1$和$G_2$的不交并可能不是$k$可公平选择的.给定任何$m \in \mathbb{N}$，对于其$K_{1,m}$是公平地可$k$选择的$k$的值是已知的。此外，公平$2$ -choosable图的完整表征是未知的。考虑到这些事实，我们研究了$\sum_{i=1}^n K_{1,m_i}$的公平选择性，$n$星的不相交联合。我们表明，确定是否$\sum_{i=1}^n K_{1,m_i}$是公平的选择是NP-完全的两种颜色的相同的列表时，分配给每个顶点。我们完全确定当两颗星（或$n \geq 2$相同的明星）的不相交工会是公平的2-choosable，我们目前的结果公平的$k$ -任意$k$的两颗星的不相交工会的可选择性。
英文摘要：
Equitable $k$-choosability is a list analogue of equitable $k$-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs $G_1$ and $G_2$ are equitably $k$-choosable, the disjoint union of $G_1$ and $G_2$ may not be equitably $k$-choosable. Given any $m \in \mathbb{N}$ the values of $k$ for which $K_{1,m}$ is equitably $k$-choosable are known. Also, a complete characterization of equitably $2$-choosable graphs is not known. With these facts in mind, we study the equitable choosability of $\sum_{i=1}^n K_{1,m_i}$, the disjoint union of $n$ stars. We show that determining whether $\sum_{i=1}^n K_{1,m_i}$ is equitably choosable is NP-complete when the same list of two colors is assigned to every vertex. We completely determine when the disjoint union of two stars (or $n \geq 2$ identical stars) is equitably 2-choosable, and we present results on the equitable $k$-choosability of the disjoint union of two stars for arbitrary $k$.
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