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Sep 6, 2010 at 21:12 comment added Andrew D. King In that case I would consider how to ask the question in terms of hypergraphs and ask a question in this context. I'm confident somebody will know a better bound than $\Delta+1$. See, for example, this conjecture: garden.irmacs.sfu.ca/?q=op/a_generalization_of_vizings_theorem
Sep 6, 2010 at 13:04 comment added Niel de Beaudrap The setting which I'm looking at puts on very few obvious restrictions on the class of graphs I'm looking at. I'm in fact looking at the generalization of the line graph to hypergraphs; so what I'm after would actually be something like the hyper-edge-chromatic number of a hypergraph. I may ask a separate question about that soon.
Sep 4, 2010 at 15:44 history answered Andrew D. King CC BY-SA 2.5