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Circular coloring of signed graphs. (English) Zbl 1383.05103
The authors have done a very good work of generalizing the concept of \((k, d)\) colorings, and the circular chromatic number to signed graphs. While establishing certain basic facts about these colorings they also analyzed the differences to these results for unsigned graphs also. It is remarkable that they brought out the difference between the circular chromatic number and the chromatic number of signed graphs as at most 1 and for showing the existence of signed graphs where such a difference is exactly 1. They have also given a nice characterization for a signed graph on \(n\) vertices if such a difference is less than 1. Moreover stimulated by the exhaustive work of the famous Xuding Zhu on circular colorings of graphs the authors have demonstrated the equivalence of \((k, d)\) colorings to \(r\)-colorings on signed graphs. They also remarked from one of their results in the paper that for a signed bipartite graph the chromatic spectrum starts from 2 and so is the circular chromatic spectrum and opined that it is worth to bring out the circular chromatic spectrum of signed graphs.

05C15 Coloring of graphs and hypergraphs
05C22 Signed and weighted graphs
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