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Mixed Graph Colorings: A Historical Review

United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova Street 6, 220012 Minsk, Belarus
Mathematics 2020, 8(3), 385; https://doi.org/10.3390/math8030385
Received: 31 January 2020 / Revised: 1 March 2020 / Accepted: 2 March 2020 / Published: 9 March 2020
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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. View Full-Text
Keywords: mixed graph; vertex coloring; chromatic number; edge coloring; chromatic index; chromatic polynomial; unit-time scheduling; makespan criterion mixed graph; vertex coloring; chromatic number; edge coloring; chromatic index; chromatic polynomial; unit-time scheduling; makespan criterion
MDPI and ACS Style

Sotskov, Y.N. Mixed Graph Colorings: A Historical Review. Mathematics 2020, 8, 385.

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