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 five, or contains a connected component with two vertices of degree four. We also give three conditions on a forest H of maximum degree at most four and with at most one vertex of degree four 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 four.

Citation

Golovach, P. A., Paulusma, D., & Ries, B. (2012, December). Coloring graphs characterized by a forbidden subgraph

• Presentation Conference Type: Conference Paper (published)
• Publication Date: 2012
• Deposit Date: Mar 11, 2013
• Print ISSN: 0302-9743
• Volume: 7464
• Pages: 443-454
• Series Title: Lecture notes in computer science
• Series Number: 7464
• Series ISSN: 0302-9743,1611-3349
• Book Title: Mathematical foundations of computer science 2012 : 37th International Symposium, MFCS 2012, Bratislava, Slovakia, 27-31 August 2012 ; proceedings.
• ISBN: 9783642325885
• DOI: https://doi.org/10.1007/978-3-642-32589-2_40
• Public URL: https://durham-repository.worktribe.com/output/1156276
• Additional Information: Series: Lecture Notes in Computer Science, Volume 7464