On a Conjecture for a Hypergraph Edge Coloring Problem
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Let H =(β³βͺπ₯ ,E βͺβ°) be a hypergraph with two hypervertices π’_1 and π’_2 where β³ =π’_1βͺπ’_2 and π’_1β©π’_2 =β . An edge {h ,j}β E in a bi-partite multigraph graph (β³βͺπ₯ ,E) has an integer multiplicity b_j h, and a hyperedge {π’_β ,j}ββ°, β=1,2, has an integer multiplicity a_j β. It has been conjectured in [5] that Ο' (H) =βΟ' _f (H)β, where Ο' (H) and Ο' _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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