What are the best results for upper bounds on the edge-chromatic χ'(H) number of k-uniform hypergraphs H?
- A k-uniform hypergraph is one in which every edge contains precisely k vertices.
- A proper edge colouring, as usual, is one in which edges of the same colour are disjoint.
I'm actually interested in principle in results for labelled k-uniform hypergraphs, where two edges may actually have the same edge set; but I would also be interested in results where edges must have distinct vertex-sets.
There seem to be several results around with added restrictions (the number of edges being small; the intersection size of edges being bounded above by a constant, etc.), but the only additional restrictions I would be interested in would be degree bounds, e.g. Δ(H) < f(k) for some strictly monotonic function f. (Equivalently, taking the dual hypergraph, I would also be interested in results for k-regular hypergraphs, but which are not necessarily uniform. Perhaps there are monotonically increasing bounds on the cardinality of the edges in terms of other parameters, but there are not other additional restrictions.)
In this setting, what are the best known upper bounds on χ'(H), expressed in terms of the hypergraph size (the number of vertices and/or edges), the maximum degree, and other such basic structural properties?
I would be especially interested in constructive proofs (i.e. ones which describe algorithms, or at least randomized constructions with high probability of success).