Given an undirected graph, a conflict-free coloring (CFON*) is an assignment of colors to a subset of the vertices of the graph such that for every vertex there exists a color that is assigned to exactly one vertex in its open neighborhood. The minimum number of colors required for such a coloring is called the conflict-free chromatic number. The decision version of the CFON* problem is NP-complete even on planar graphs. In this paper, we show the following results. The CFON* problem is fixed-parameter tractable with respect to the combined parameters clique width and the solution size. We study the problem on block graphs and cographs, which have bounded clique width. For both graph classes, we give tight bounds of three and two respectively for the CFON* chromatic number. We study the problem on the following intersection graphs: interval graphs, unit square graphs and unit disk graphs. We give tight bounds of two and three for the CFON* chromatic number for proper interval graphs and interval graphs. Moreover, we give upper bounds for the CFON* chromatic number on unit square and unit disk graphs. We also study the problem on split graphs and Kneser graphs. For split graphs, we show that the problem is NP-complete. For Kneser graphs $K(n,k)$, when $n\geq k(k+1)^2 + 1$, we show that the CFON* chromatic number is $k+1$. We also study the closed neighborhood variant of the problem denoted by CFCN*, and obtain analogous results in some of the above cases.
翻译:在未定向的图表中,无冲突色素(CFON*)是将颜色分配给图中一组脊椎的分层,这样每个顶端都有一个完全指定在开放区区区一个顶端的颜色。这种颜色所需的最低颜色数量称为无冲突色谱数字。CFON* 问题的决定版本甚至用平面图也完整。在本文中,我们展示了以下结果。CFON* 问题是固定的参数,对于总值的球质宽度和解决方案大小,每个顶端都有固定的参数。我们研究了块形图和cograph的颜色。对于这两个类别,我们分别给出了3个和2个不同的颜色。我们研究了以下交叉图的问题:间图、单位正方位正方图和单位磁盘图。我们用2个和3个基数的基数的基质参数,我们用CFON* 平面图和基数的基面图和基面图的基数也展示了1号。我们用平面图的上,我们用平面图和基面图的基数的基数的基数,我们做了一个基数研究。