3 research outputs found
3-List Colouring Permutation Graphs
3-list colouring is an NP-complete decision problem. It is hard even on
planar bipartite graphs. We give a polynomial-time algorithm for solving 3-list
colouring on permutation graphs
On List Colouring and List Homomorphism of Permutation and Interval Graphs
List colouring is an NP-complete decision problem even if the total number of
colours is three. It is hard even on planar bipartite graphs. We give a
polynomial-time algorithm for solving list colouring of permutation graphs with
a bounded total number of colours. More generally we give a polynomial-time
algorithm that solves the list-homomorphism problem to any fixed target graph
for a large class of input graphs including all permutation and interval
graphs
Coloring Problems on Bipartite Graphs of Small Diameter
We investigate a number of coloring problems restricted to bipartite graphs
with bounded diameter. We prove that the -List Coloring, List -Coloring,
and -Precoloring Extension problems are NP-complete on bipartite graphs with
diameter at most , for every and every , and for and
, and that List -Coloring is polynomial when (i.e., on
complete bipartite graphs) for every . Since -List Coloring was
already known to be NP-complete on complete bipartite graphs, and polynomial
for on general graphs, the only remaining open problems are List
-Coloring and -Precoloring Extension when .
We also prove that the Surjective -Homomorphism problem is NP-complete
on bipartite graphs with diameter at most , answering a question posed by
Bodirsky, K\'ara, and Martin [Discret. Appl. Math. 2012]. As a byproduct, we
get that deciding whether can be partitioned into 3 subsets each
inducing a complete bipartite graph is NP-complete. An attempt to prove this
result was presented by Fleischner, Mujuni, Paulusma, and Szeider [Theor.
Comput. Sci. 2009], but we realized that there was an apparently non-fixable
flaw in their proof.
Finally, we prove that the -Fall Coloring problem is NP-complete on
bipartite graphs with diameter at most , and give a polynomial reduction
from -Fall Coloring on bipartite graphs with diameter to -Precoloring
Extension on bipartite graphs with diameter . The latter result implies that
if -Fall Coloring is NP-complete on these graphs, then the complexity gaps
mentioned above for List -Coloring and -Precoloring Extension would be
closed. This would also answer a question posed by Kratochv\'il, Tuza, and
Voigt [Proc. of WG 2002].Comment: 21 pages, 9 figure