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Coloring graphs characterized by a forbidden subgraph.

Golovach, P.A. and Paulusma, D. and Ries, B. (2015) 'Coloring graphs characterized by a forbidden subgraph.', Discrete applied mathematics., 180 . pp. 101-110.


The Coloring problem is to test whether a given graph can be colored with at most k colors for some given k, such that no two adjacent vertices receive the same color. The complexity of this problem on graphs that do not contain some graph H as an induced subgraph is known for each fixed graph H. A natural variant is to forbid a graph H only as a subgraph. We call such graphs strongly H-free and initiate a complexity classification of Coloring for strongly H-free graphs. We show that Coloring is NP-complete for strongly H-free graphs, even for k=3, when H contains a cycle, has maximum degree at least 5, or contains a connected component with two vertices of degree 4. We also give three conditions on a forest H of maximum degree at most 4 and with at most one vertex of degree 4 in each of its connected components, such that Coloring is NP-complete for strongly H-free graphs even for k=3. Finally, we classify the computational complexity of Coloring on strongly H-free graphs for all fixed graphs H up to seven vertices. In particular, we show that Coloring is polynomial-time solvable when H is a forest that has at most seven vertices and maximum degree at most 4.

Item Type:Article
Keywords:Complexity, Algorithms, Graph coloring, Forbidden subgraphs.
Full text:(AM) Accepted Manuscript
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Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in Discrete applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete applied mathematics, 180, 2015, 10.1016/j.dam.2014.08.008
Date accepted:No date available
Date deposited:06 January 2015
Date of first online publication:January 2015
Date first made open access:No date available

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