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arXiv · 2006.06393

On a Conjecture for a Hypergraph Edge Coloring Problem

Wieslaw Kubiak
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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.

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Wieslaw Kubiak. 2020-06-11. On a Conjecture for a Hypergraph Edge Coloring Problem. https://arxiv.org/abs/2006.06393

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