On a Conjecture for a Hypergraph Edge Coloring Problem
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Let H =(ℳ∪𝒥 ,E ∪ℰ) be a hypergraph with two hypervertices 𝒢_1 and 𝒢_2 where ℳ =𝒢_1∪𝒢_2 and 𝒢_1∩𝒢_2 =∅. An edge {h ,j}∈ E in a bi-partite multigraph graph (ℳ∪𝒥 ,E) has an integer multiplicity b_j h, and a hyperedge {𝒢_ℓ ,j}∈ℰ, ℓ=1,2, has an integer multiplicity a_j ℓ. It has been conjectured in [5] that χ' (H) =⌈χ' _f (H)⌉, where χ' (H) and χ' _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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