The complexity of colouring problems on dense graphs
Theoretical Computer Science
The complexity of coloring graphs without long induced paths
Acta Cybernetica
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Complexity of Coloring Graphs without Forbidden Induced Subgraphs
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
The 3-Colorability Problem on Graphs with Maximum Degree Four
SIAM Journal on Computing
3-Colorability ∈ P for P6-free graphs
Discrete Applied Mathematics - The 1st cologne-twente workshop on graphs and combinatorial optimization (CTW 2001)
Vertex Colouring and Forbidden Subgraphs – A Survey
Graphs and Combinatorics
Note: On the complexity of 4-coloring graphs without long induced paths
Theoretical Computer Science
Three Complexity Results on Coloring Pk-Free Graphs
Combinatorial Algorithms
A Certifying Algorithm for 3-Colorability of P5-Free Graphs
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Deciding k-Colorability of P 5-Free Graphs in Polynomial Time
Algorithmica - Including a Special Section on Genetic and Evolutionary Computation; Guest Editors: Benjamin Doerr, Frank Neumann and Ingo Wegener
Colouring vertices of triangle-free graphs
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Theoretical Computer Science
Coloring graphs without short cycles and long induced paths
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
On the parameterized complexity of coloring graphs in the absence of a linear forest
Journal of Discrete Algorithms
Linear time algorithm for computing a small biclique in graphs without long induced paths
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Coloring graphs characterized by a forbidden subgraph
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
4-coloring H-free graphs when H is small
Discrete Applied Mathematics
Three complexity results on coloring Pk-free graphs
European Journal of Combinatorics
Information Processing Letters
Colouring of graphs with Ramsey-type forbidden subgraphs
Theoretical Computer Science
List coloring in the absence of two subgraphs
Discrete Applied Mathematics
Coloring graphs without short cycles and long induced paths
Discrete Applied Mathematics
| Hi-index | 5.23 |
A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph P"k denotes a path on k vertices. The @?-Coloring problem is the problem to decide whether a graph can be colored with at most @? colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P"8-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P"9-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NP-complete for P"8-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P"7-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P"2+P"4)-free graphs, a subclass of P"7-free graphs. Here P"2+P"4 denotes the disjoint union of a P"2 and a P"4. We denote the disjoint union of s copies of a P"3 by sP"3 and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP"3-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring for H-free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.