Narrowing down the gap on the complexity of coloring Pk-free graphs
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Colouring vertices of triangle-free graphs
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Coloring graphs without short cycles and long induced paths
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Updating the complexity status of coloring graphs without a fixed induced linear forest
Theoretical Computer Science
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Theoretical Computer Science
List coloring in the absence of a linear forest
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
4-coloring h-free graphs when h is small
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
4-coloring H-free graphs when H is small
Discrete Applied Mathematics
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We prove three complexity results on vertex coloring problems restricted to P k -free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P 6-free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P 5-free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P 6-free graphs. This implies a simpler algorithm for checking the 3-colorability of P 6-free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P 7-free graphs. This problem was known to be polynomially solvable for P 5-free graphs and NP-complete for P 8-free graphs, so there remains one open case.