3-Colorability ∈ P for P 6 -free graphs
Bert Randerath,Ingo Schiermeyer +1 more
- Vol. 136, Iss: 2, pp 299-313
TLDR
In this paper, it was shown that 3-colorability can be decided in polynomial time for the class of P6-free graphs with bounded dominating subgraphs.Abstract:
In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.read more
Citations
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Journal ArticleDOI
Deciding k -Colorability of P 5 -Free Graphs in Polynomial Time
TL;DR: In this paper, it was shown that for every fixed integer k, there exists a polynomial-time algorithm for determining whether a P5-free graph admits a k-coloring, and finding one, if it does.
Journal ArticleDOI
Vertex Colouring and Forbidden Subgraphs – A Survey
Bert Randerath,Ingo Schiermeyer +1 more
TL;DR: This work surveys results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions in order to obtain useful results from a graph coloring formulation of his problem.
Journal ArticleDOI
A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs
TL;DR: In this article, the authors survey known results on the computational complexity of k-coloring and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.
Journal ArticleDOI
Improved complexity results on k -coloring P t -free graphs
TL;DR: It is proved that 4-COLORING is NP-complete for P 7 -free graphs, and that 5-COLORing is NP -complete forP 6 -free graph, and the second result is the first NP-hardness result for k -COLORING P 6 - free graphs.
Journal ArticleDOI
Three-coloring and list three-coloring of graphs without induced paths on seven vertices
TL;DR: A polynomial time algorithm is presented that determines if an input graph containing no induced seven-vertex path is 3-colorable, which affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002.
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