
A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter
In a graph, a matching cut is an edge cut that is a matching. Matching C...
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On List kColoring Convex Bipartite Graphs
List kColoring (Li kCol) is the decision problem asking if a given gra...
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Complexity of C_kcoloring in hereditary classes of graphs
For a graph F, a graph G is Ffree if it does not contain an induced sub...
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Sparsification Lower Bounds for List HColoring
We investigate the List HColoring problem, the generalization of graph ...
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Complexity of fall coloring for restricted graph classes
We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), ...
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Notes on complexity of packing coloring
A packing kcoloring for some integer k of a graph G=(V,E) is a mapping ...
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Faster 3coloring of smalldiameter graphs
We study the 3Coloring problem in graphs with small diameter. In 2013, ...
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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 kList Coloring, List kColoring, and kPrecoloring Extension problems are NPcomplete on bipartite graphs with diameter at most d, for every k≥ 4 and every d≥ 3, and for k=3 and d≥ 4, and that List kColoring is polynomial when d=2 (i.e., on complete bipartite graphs) for every k ≥ 3. Since kList Coloring was already known to be NPcomplete on complete bipartite graphs, and polynomial for k=2 on general graphs, the only remaining open problems are List 3Coloring and 3Precoloring Extension when d=3. We also prove that the Surjective C_6Homomorphism problem is NPcomplete on bipartite graphs with diameter at most 4, answering a question posed by Bodirsky, Kára, and Martin [Discret. Appl. Math. 2012]. As a byproduct, we get that deciding whether V(G) can be partitioned into 3 subsets each inducing a complete bipartite graph is NPcomplete. 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 nonfixable flaw in their proof. Finally, we prove that the 3Fall Coloring problem is NPcomplete on bipartite graphs with diameter at most 4, and give a polynomial reduction from 3Fall Coloring on bipartite graphs with diameter 3 to 3Precoloring Extension on bipartite graphs with diameter 3. The latter result implies that if 3Fall Coloring is NPcomplete on these graphs, then the complexity gaps mentioned above for List kColoring and kPrecoloring Extension would be closed. This would also answer a question posed by Kratochvíl, Tuza, and Voigt [Proc. of WG 2002].
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