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Colouring Diamond-free Graphs

Authors Konrad K. Dabrowski, François Dross, Daniël Paulusma

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Keywords
  • colouring
  • clique-width
  • diamond-free
  • graph class
  • hereditary graph class

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Abstract

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

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Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring Diamond-free Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.16

Author Details

Konrad K. Dabrowski
François Dross
Daniël Paulusma

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