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Circular coloring of signed graphs

Let k,d ( 2d≤k) be two positive integers. We generalize the well‐studied notions of (k,d) ‐colorings and of the circular chromatic number χc to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number χ. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on n vertices, if the difference is smaller than 1, then there exists εn>0 , such that the difference is at most 1−εn .