三次图的冠积的均匀染色比一般染色困难
Equitable coloring of corona products of cubic graphs is harder than ordinary coloring
摘要：
一个图是公平$k$ -可着色的，如果它的顶点可以被划分成$k$独立的集合，使得任意两个集合中的顶点数相差至多1。存在这种着色的最小$k$被称为$G$的\emph{equitable chromatic number}，并且它被表示为$χ_{=}(G)$。本文研究了三次图的冠的$χ_=$的确定问题。虽然三次图的冠的普通着色问题在多项式时间内是可解的，但对这类图的均匀着色问题则是NP-难的。我们提供了多项式可解的情况下，冠的三次图，并证明了在一般情况下的NP-硬度。作为一个副产品，我们得到了一个简单的线性时间算法的公平着色，使用$χ_=(G)$或$χ_=(G)+1$颜色的图。我们的算法是最好的，除非$P=NP$。因此，三次冠似乎是唯一已知的一类图的公平着色是困难的比普通着色。
英文摘要：
A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the \emph{equitable chromatic number} of $G$ and it is denoted by $χ_{=}(G)$. In this paper the problem of determinig $χ_=$ for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses $χ_=(G)$ or $χ_=(G)+1$ colors. Our algorithm is best possible, unless $P=NP$. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.
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