TY - JOUR
T1 - On the complexity of some colorful problems parameterized by treewidth
AU - Fellows, Michael R.
AU - Fomin,, Fedor V.
AU - Lokshtanov, Daniel
AU - Rosamond, Frances
AU - Saurabh, Sakel
AU - Szeider, Stefan
AU - Thomassen, Carsten
PY - 2011
Y1 - 2011
N2 - In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph.1.The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C"v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C"v, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth. 2.An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterized by the number of colors on the lists. 3.The list chromatic number@g"l(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L"v of colors, where each list has length at least r, there is a choice of one color from each vertex list L"v yielding a proper coloring of G. We show that the problem of determining whether @g"l(G)=
AB - In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph.1.The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C"v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C"v, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth. 2.An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterized by the number of colors on the lists. 3.The list chromatic number@g"l(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L"v of colors, where each list has length at least r, there is a choice of one color from each vertex list L"v yielding a proper coloring of G. We show that the problem of determining whether @g"l(G)=
U2 - 10.1016/j.ic.2010.11.026
DO - 10.1016/j.ic.2010.11.026
M3 - Journal article
VL - 209
SP - 143
EP - 153
JO - Information and Computation
JF - Information and Computation
SN - 0890-5401
IS - 2
ER -