The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
On the NP-completeness of the k-colorability problem for triangle-free graphs
Discrete Mathematics
Generalized coloring for tree-like 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
Complexity of Coloring Graphs without Forbidden Induced Subgraphs
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
Triangle-free graphs and forbidden subgraphs
Discrete Applied Mathematics - Sixth Twente Workshop on Graphs and Combinatorial Optimization
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of 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
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
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
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
| Hi-index | 0.04 |
For an integer k=1, a graph G is k-colorable if there exists a mapping c:V"G-{1,...,k} such that c(u)c(v) whenever u and v are two adjacent vertices. For a fixed integer k=1, the k-Coloring problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g=4 we determine a lower bound @?(g), such that every graph with girth at least g and with no induced path on @?(g) vertices is 3-colorable. We also show that for all fixed integers k,@?=1, thek-Coloring problem can be solved in polynomial time for graphs with no induced cycle on four vertices and no induced path on @? vertices. As a consequence, for all fixed integers k,@?=1 and g=5, the k-Coloring problem can be solved in polynomial time for graphs with girth at least g and with no induced path on @? vertices. This result is best possible, as we prove the existence of an integer @?^*, such that already 4-Coloring is NP-complete for graphs with girth 4 and with no induced path on @?^* vertices.