Note: On the complexity of 4-coloring graphs without long induced paths

Authors:
Van Bang Le;Bert Randerath;Ingo Schiermeyer
Affiliations:
Institut für Informatik, Universität Rostock, D-18051 Rostock, Germany;Institut für Informatik, Universität zu Köln, D-50969 Köln, Germany;Institut für Diskrete Mathematik und Algebra, TU Bergakademie Freiberg, D-09596 Freiberg, Germany
Venue:
Theoretical Computer Science
Year:
2007

Citing 8
Cited 13

Weighted parameters in (P5,&Pmacr;5)-free graphs

Discrete Applied Mathematics
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
Three-colourbility and forbidden subgraphs. II: polynomial algorithms

Discrete Mathematics
Some results on graphs without long induced paths

Information Processing Letters
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
Recognizing Berge Graphs

Combinatorica

A Note on k-Colorability of P5-Free Graphs

MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
A Certifying Algorithm for 3-Colorability of P5-Free Graphs

ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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
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
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
4-coloring H-free graphs when H is small

Discrete Applied Mathematics
Three complexity results on coloring Pk-free graphs

European Journal of Combinatorics
Coloring graphs without short cycles and long induced paths

Discrete Applied Mathematics

Quantified Score

Hi-index	5.23

Visualization

Abstract

We show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an NP-complete problem, improving a previous NP-completeness result proved by Woeginger and Sgall in 2001. The complexity of 4-coloring graphs without induced paths on five vertices remains open. We show that deciding if a graph without induced paths or cycles on five vertices can be colored with 4 colors can be done in polynomial time. A step in our algorithm uses the well-known and deep fact due to Grotschel, Lovasz and Schrijver stating that perfect graphs can be optimally colored in polynomial time.