论文标题
适当的无冲突$ k $ - 颜色问题和奇数$ k $ - 颜色的问题在两部分图表上是NP的完整问题
The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs
论文作者
论文摘要
如果每个非分离顶点$ v $都有一个邻居，其颜色在$ v $的附近是唯一的，则图形的适当着色是\ emph {适当的无冲突}。图形的适当着色是\ emph {奇数}，如果对于每个非分离顶点$ v $，则在$ v $附近的颜色出现奇数次。对于整数$ k $，\ textsc {pcf $ k $ -coloring}问题询问输入图是否承认适当的无冲突$ k $ - 颜色和\ textsc {奇数$ k $ -coloring}询问输入图是否接收奇数$ k $ -coloring。我们表明，对于每个整数$ k \ geq3 $，即使输入图是双分部分，这两个问题都是NP的。此外，我们表明\ textsc {pcf $ 4 $ - 颜色}问题是当输入图是平面时NP complete。
A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring} problem asks whether an input graph admits a proper conflict-free $k$-coloring and the \textsc{Odd $k$-Coloring} asks whether an input graph admits an odd $k$-coloring. We show that for every integer $k\geq3$, both problems are NP-complete, even if the input graph is bipartite. Furthermore, we show that the \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph is planar.
