# A generalization of Vizing's Theorem?

\begin{conjecture} Let $H$ be a simple $d$-\Def[uniform]{hypergraph} hypergraph, and assume that every set of $d-1$ points is contained in at most $r$ edges. Then there exists an $r+d-1$-edge-coloring so that any two edges which share $d-1$ vertices have distinct colors. \end{conjecture}

Vizing's Theorem is equivalent to the above statement for $d=2$. 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.

## Reference

Could someone please add a reference? There should be some paper (or a conference talk?) where Rosenfeld proposed the conjecture.

-DOT