On a Conjecture for a Hypergraph Edge Coloring Problem

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 $\mathrm{.}Anedge${h ,j} \in E$ in a bi-partite multigraph graph $(\mathcal{M}\cup \mathcal{J},E)$ has an integer multiplicity $b_{jh}$, and a hyperedge ${$\mathcal{G}${\ell } ,j} \in $\mathcal{E}$$,$\ell=1,2$,hasanintegermultiplicity$a{j \ell }$. 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.