Let 2P3 denote the disjoint union of two paths on three vertices. A graph G that has no subgraph isomorphic to a graph H is called H-free. The Vertex Coloring problem is the problem to determine the chromatic number of a graph. Its computational complexity for triangle-free H-free graphs has been classified for every fixed graph H on at most 6 vertices except for the case H=2P3. This remaining case is posed as an open problem by Dabrowski, Lozin, Raman and Ries. We solve their open problem by showing polynomial-time solvability.
- Graph coloring
- Forbidden subgraphs
Broersma, H. J., Golovach, P. A., Paulusma, D., & Song, J. (2012). Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time. Theoretical computer science, 423, 1-10. https://doi.org/10.1016/j.tcs.2011.12.076