List-coloring – Parameterizing from triviality

Abstract

The classical graph coloring problem is given an undirected graph and the goal is to color the vertices of the graph with the minimum number of colors so that endpoints of each edge gets different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored. Most versions of the problems are hard in several paradigms including approximation and parameterized complexity. We consider a few versions of the problems that are polynomial time solvable and try to extend the notion of feasible algorithms by parameterizing suitably in the paradigm of parameterized complexity. In particular, we provide extensions of these polynomial time solvable coloring problems that belong to the extreme ends of the parameterized complexity hierarchy including fixed-parameter tractable (FPT) class or the complexity class XP or W[1]-hard or para-NP -hard (i.e. NP -hard even for constant values of the parameter).Specifically, we consider generalizations of odd cycle transversal and edge bipartization in the context of list-coloring and show them fixed-parameter tractable. We also look at list coloring where each list has n−k colors and the goal is to color the vertices respecting the list. We show that the problem is in the parameterized complexity class XP, when parameterized by k.