We consider the list-coloring problem from the perspective of parameterized complexity. In the classical graph coloring problem we are given an undirected graph and the goal is to color the vertices of the graph with minimum number of colors so that end points of each edge get different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored. An interesting parameterization for graph coloring that has been studied is whether the graph can be colored with n − k colors, where k is the parameter and n is the number of vertices. This is known to be fixed parameter tractable. Our main result is that this can be generalized for list-coloring as well. More specifically, we show that, given a graph with each vertex having a list of size n − k, it can be determined in \(f(k)n^{{\mathcal O}(1)}\) time, for some function f of k, whether there is a coloring that respects the lists.

我们从参数化复杂度的角度考虑列表着色问题。在经典图着色问题中，我们给了无向图，目标是用最少的颜色数着色图的顶点，以使每个边缘的端点得到不同的颜色。在列表着色中，每个顶点都有一个允许的颜色列表，可以对其进行着色。研究过的有趣的图形着色参数化是图形是否可以用n - k种颜色着色，其中k是参数，n是是顶点数。已知这是固定参数可处理的。我们的主要结果是，这也可以推广到列表着色。更具体地说，我们表明，给定一个图，每个图的顶点大小为n - k，则可以在\（f（k）n ^ {{\\ mathcal O}（1）} \）的时间内确定，一些功能˚F的ķ，是否有着色尊重列表。