# Bounds on chromatic index

Let $H$ be a hypergraph of maximum vertex-degree $\Delta$. (That is, for all vertices $x$, we have $| \{ e \in H \mid x \in e \} | \leq \Delta$) Are there any bounds on the chromatic index $\chi_e(H)$ of the form $$\chi_e(H) \leq c \Delta$$ for some constant $c$?

If not, are there any simple criteria on $H$ that can guarantee this?

Note that if $H$ is a multi-graph, then this follows from Shannon's theorem.

• Perhaps you know of Alon&Kim, "A note on the degree, size and chromatic index of a uniform hypergraph"? They offer a conjecture that depends on a parameter (the number of shared vertices of any two edges) times $\Delta$, which might suggest there is no constant. (PDF paper download.) – Joseph O'Rourke Dec 30 '15 at 19:20

The answer is definitely no when $c$ is supposed to be independent of number of vertices and size of edges. You can easily construct hypergraphs with an arbitrary number of pairwise intersecting edges and maximum degree $2$.