Some (in)tractable Parameterizations of Coloring and List-Coloring

Abstract

Graph Coloring and its generalization list coloring are fundamental graph optimization problems with various applications. 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. More specifically, It is known that given a planar graph with any list of size 5 for each vertex, there is a proper coloring of the graph such that each vertex gets its color from its list. We show that if the graph is k vertices away from a planar graph, then deciding whether such a coloring exists is para-NP-hard when parameterized by k, i.e. it is NP-hard for even constant values of k. It is known that any graph with maximum degree 3 is 3-colorable unless the graph is a 4-clique. We show that if the graph is k vertices away from a maximum degree 3 graph, then determining whether it is 3-colorable is para-NP-hard when parameterized by k. It is known that if each vertex has a list of size 2, then the list coloring which asks whether there is a coloring respecting the lists is polynomial time solvable. We show that if only k vertices have lists of size more than 2, then the problem becomes W[1]-hard. It is known that determining whether a graph on n vertices is \(n-k\) colorable, is fixed-parameter tractable on k. We consider the list coloring variation of it where each vertex has a list of size \(n-k\) and we ask whether the graph has a coloring respecting the lists of colors. We show that the problem has an XP algorithm, i.e. an algorithm with runtime \(n^{O(k)}\). At least this shows that the problem cannot be para-NP-hard unless \(P =NP\). We leave open the question whether the problem is fixed-parameter tractable. Finally, it is known that \(2-\) List coloring is polynomial time solvable. If there is no such coloring, then we address the following natural question: are there k vertices or edges whose removal results in a feasible coloring. We show that these versions are fixed-parameter tractable when parameterized by k. These generalize the odd cycle transversal problem and edge-bipartization problem which are well-studied problems particularly in parameterized complexity.