Hypergraph chromatic index (${\chi }^{\prime }(H)$)

The chromatic index of a hypergraph is the minimum number of colours needed to colour its edges so that intersecting edges have different colours. Each colour class is a matching in a hypergraph. It is at least the maximum hypergraph vertex degree, and at most $r(\Delta -1)+1$ for an $r$-uniform hypergraph of maximum degree $\Delta$, by greedily colouring edges: an edge meets at most $r(\Delta -1)$ others.

Given an $r$-uniform hypergraph $H$ and $K$ colours, construct an $(r+1)$-uniform auxiliary hypergraph with one task vertex for every $e\in E(H)$ and one resource vertex $(v,c)$ for every original vertex and colour. The assignment edge for $(e,c)$ contains the task $e$ and all resources $(v,c)$ with $v\in e$. A matching in a hypergraph of assignments is exactly a proper partial edge colouring. Task degrees are $K$; resource degrees are $d_H(v)$. Pairwise auxiliary hypergraph codegrees are at most $\max (1,{\max }_{v\ne w}{d}_H(v,w))$. Thus nearly equal original degrees and small original hypergraph codegrees transfer to the setting of a Rödl nibble. A locally balanced Rödl nibble must control the unmatched task count in every original star, not merely the total uncovered proportion.