On a Conjecture for a Hypergraph Edge Coloring Problem

TL;DR

This paper proves a conjecture relating the edge chromatic number and fractional edge chromatic number in a specific hypergraph class, with implications for scheduling and timetabling problems.

Contribution

The paper provides a proof of a conjecture connecting edge coloring and fractional edge coloring in a particular hypergraph model.

Findings

Proves the conjecture that hi' (H) = eil(hi'_{f} (H)) in the specified hypergraph.

Establishes a theoretical foundation for hypergraph edge coloring in scheduling applications.

Links hypergraph coloring theory to practical problems like timetabling and multiprocessor scheduling.

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.