Colouring Diamond-free Graphs
Konrad K. Dabrowski, Fran\c{c}ois Dross, Dani\"el Paulusma

TL;DR
This paper classifies the computational complexity of graph coloring for specific classes of diamond-free graphs, introduces techniques to prove bounded clique-width, and advances understanding of graph classes with bounded clique-width.
Contribution
It provides a complete classification for (diamond, H)-free graphs with up to five vertices and introduces a new reduction technique for proving bounded clique-width.
Findings
Classified complexity for all (diamond, H)-free graphs with |V(H)| ≤ 5.
Proved bounded clique-width for (diamond, P_1+2P_2)-free graphs.
Reduced open cases of bounded clique-width from 13 to 8.
Abstract
The Colouring problem is that of deciding, given a graph and an integer , whether admits a (proper) -colouring. For all graphs up to five vertices, we classify the computational complexity of Colouring for -free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for -free graphs. Our technique for handling this case is to reduce the graph under consideration to a -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 -free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of -free graphs, and our five new classes of bounded clique-width reduce the number of…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
