A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the triangles containing any given edge is one. We prove that for all ǫ > 0, every large enough graph graph on n vertices with minimum degree at least (0.9 + ǫ)n has a fractional triangle decomposition. This improves a result of Garaschuk that the same result holds for graphs with minimum degree at least 0.956n. Together with a recent result of Barber, Kühn, Lo and Osthus, this implies that for all ǫ > 0, every large enough triangle divisible graph on n vertices with minimum degree at least (0.9+ǫ)n admits a triangle decomposition.
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, P1 + 2P2)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 (H1, H2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H1, H2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8. ⋆ First and last author supported by EPSRC (EP/K025090/1). An extended abstract of this paper appeared in the proceedings of SWAT 2016 [17].
An (F, F d )-partition of a graph is a vertex-partition into two sets F and F d such that the graph induced by F is a forest and the one induced by F d is a forest with maximum degree at most d. We prove that every triangle-free planar graph admits an (F, F5)-partition. Moreover we show that if for some integer d there exists a trianglefree planar graph that does not admit an (F, F d )-partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.