A generalization of Vizing's Theorem?
Conjecture Let be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors.
Vizing's Theorem is equivalent to the above statement for . For higher dimensions, this problem looks difficult since the main tool used in the proof of Vizing's theorem (Kempe chains) do not appear to work.