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Coloring graphs characterized by a forbidden subgraph

Golovach, P.A.; Paulusma, D.; Ries, B.

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P.A. Golovach

B. Ries


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.


Golovach, P., Paulusma, D., & Ries, B. (2015). Coloring graphs characterized by a forbidden subgraph. Discrete Applied Mathematics, 180, 101-110.

Journal Article Type Article
Publication Date Jan 1, 2015
Deposit Date Dec 20, 2014
Publicly Available Date Jan 6, 2015
Journal Discrete Applied Mathematics
Print ISSN 0166-218X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 180
Pages 101-110
Keywords Complexity, Algorithms, Graph coloring, Forbidden subgraphs.


Accepted Journal Article (385 Kb)

Copyright 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

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