Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.Comment: Full version of ESA 201

Max-Leaves Spanning Tree is APX-hard for Cubic Graphs

We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-hardness of the related problem Minimum Connected Dominating Set for cubic graphs follows

Ramsey numbers R(K3,G) for graphs of order 10

In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.Comment: 24 pages, submitted for publication; added some comment