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.
Categories: cs.DM math.CO