On a Conjecture for a Hypergraph Edge Coloring Problem (2006.06393v1)

Abstract: 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.