  Let $H =(\mathcal{M} \cup \mathcal{J} ,E \cup \mathcal{E})$ be a hypergraph
with two hypervertices $\mathcal{G}_1$ and $\mathcal{G}_2$ where $\mathcal{M}
=\mathcal{G}_{1} \cup \mathcal{G}_{2}$ and $\mathcal{G}_{1} \cap
\mathcal{G}_{2} =\varnothing $. An edge $\{h ,j\} \in E$ in a bi-partite
multigraph graph $(\mathcal{M} \cup \mathcal{J} ,E)$ has an integer
multiplicity $b_{j h}$, and a hyperedge $\{\mathcal{G}_{\ell } ,j\} \in
\mathcal{E}$, $\ell=1,2$, has an integer multiplicity $a_{j \ell }$. It has
been conjectured in [5] that $\chi \prime (H) =\lceil \chi \prime _{f}
(H)\rceil $, where $\chi \prime (H)$ and $\chi \prime _{f} (H)$ are the edge
chromatic number of $H$ and the fractional edge chromatic number of $H$
respectively. Motivation to study this hyperedge coloring conjecture comes from
the University timetabling, and open shop scheduling with multiprocessors. We
prove this conjecture in this paper.
