Mixed graph colouring as scheduling multi-processor tasks with equal processing times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., an assignment integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, νn}, G = (V, A, E) such that if vp and vq are joined by edge [νp, νq] ∈ E their colours have be different. Further, two νp νq arc (νi, νj) colour vertex νi has no greater than νj. We prove optimal equivalent GcMPT|pi 1|Cmax finding schedule for multi-processor with unit (equal) processing times. Contrary classical shop-scheduling problems, several required process individual task in 1|Cmax. Moreover, along precedence constraints given set it subset must simultaneously. Due theorems proved this article, most analytical results been problems GcMPT |pi so far, analogous colourings graphs E), vice versa.