Coloring graphs characterized by a forbidden subgraph

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

Abstract

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.

Citation

Golovach, P., Paulusma, D., & Ries, B. (2015). Coloring graphs characterized by a forbidden subgraph. Discrete Applied Mathematics, 180, 101-110. https://doi.org/10.1016/j.dam.2014.08.008

• 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
• Electronic ISSN: 1872-6771
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 180
• Pages: 101-110
• DOI: https://doi.org/10.1016/j.dam.2014.08.008
• Keywords: Complexity, Algorithms, Graph coloring, Forbidden subgraphs.
• Public URL: https://durham-repository.worktribe.com/output/1415033