Abstract
This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced.
Highlights
Let G = (V, A, E) denote a finite mixed graph with a non-empty set V = {v1, v2, . . . , vn } of n vertices, a set A of arcs, and a set E of edges
In [39], the scheduling problem J | pi = 1|Cmax is presented as finding an optimal c< -coloring of a special mixed graph satisfying Properties 1 and 2
It is clear that equalities φk (Ck ) = Ck, Jk ∈ J imply that Fmax = Cmax and this special case G ||Cmax of the general shop scheduling problem G || Fmax is equivalent to the problem of finding an optimal c-coloring of the mixed graph G = (V, A, E), which is obtained from the weighted mixed graph Gw = (V, Aw, Ew ) as follows
Summary
In [11], a coloring of arcs and edges in the mixed graph G = (V, A, E) was determined as follows. The coloring of arcs and edges in the mixed graph G = (V, A, E) is a c< -coloring of vertices in the mixed line graph L( G ) = ( A ∪ E, A A∪E , E A∪E ), and vice versa. A c< -coloring of the vertices of the mixed line graph L( G ) is called an edge coloring of the mixed graph G = (V, A, E). An edge coloring is optimal if it uses a minimum possible number χ0 ( G ) of different colors c< (eij ) ∈ {1, 2, . Throughout the paper, we use the terminology from [12,13] for graph theory and that from [14,15] for scheduling theory
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