The complexity of the 3-colorability problem in the absence of a pair of small forbidden induced subgraphs

Malyshev, D. S.

doi:10.1016/j.disc.2015.04.019

Cited by 12 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. 5.Malyshev proved that 3‐ Coloring is

sans-serifNP

‐complete for

(C_{3}^{+ +}, \bar{C_{4} + P_{1}}, K_{1, 4})

‐free graphs after previously proving that 3‐ Coloring is

sans-serifNP

‐complete for

(C_{3}^{+ +}, K_{1, 4})

‐free graphs . Note that Theorem (i):3 already implies that 3‐ Coloring is

sans-serifNP

‐complete for

(\bar{C_{4} + P_{1}}, K_{1, 4})

‐free graphs. …”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 96%

“…Malyshev characterized exactly those pairs of graphs H 1 and H 2 each with at most five vertices for which 3‐ Coloring is polynomial‐time solvable when restricted to

(H_{1}, H_{2})

‐free graphs. His main result, obtained by combining known results with new results, can be summarized as follows.…”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 99%

“…5. Malyshev [89] proved that 3-COLOURING is NP-complete for (C ++ 3 , C 4 + P 1 , K 1,4 )-free graphs after previously proving that 3-COLOURING is NP-complete for (C ++ 3 , K 1,4 )-free graphs [88]. Note that Theorem 21 (i).3 already implies that 3-COLOURING is NP-complete for (C 4 + P 1 , K 1,4 )-free graphs.…”

Section: Results and Open Problems For (H 1 H 2 )-Free Graphsmentioning

confidence: 99%

See 2 more Smart Citations

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

131

160

View full text Add to dashboard Cite

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For a positive integer k, a k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex. Finally, we also survey results for graph classes defined by some other forbidden pattern.

show abstract

“…. 5.Malyshev proved that 3‐ Coloring is

sans-serifNP

‐complete for

(C_{3}^{+ +}, \bar{C_{4} + P_{1}}, K_{1, 4})

‐free graphs after previously proving that 3‐ Coloring is

sans-serifNP

‐complete for

(C_{3}^{+ +}, K_{1, 4})

‐free graphs . Note that Theorem (i):3 already implies that 3‐ Coloring is

sans-serifNP

‐complete for

(\bar{C_{4} + P_{1}}, K_{1, 4})

‐free graphs. …”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 96%

“…Malyshev characterized exactly those pairs of graphs H 1 and H 2 each with at most five vertices for which 3‐ Coloring is polynomial‐time solvable when restricted to