  Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well
studied notions of $(k,d)$-colorings and of the circular chromatic number
$\chi_c$ to signed graphs. This implies a new notion of colorings of signed
graphs, and the corresponding chromatic number $\chi$. 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
$\epsilon_n>0$, such that the difference is at most $1 - \epsilon_n$.
  We also show that notion of $(k,d)$-colorings is equivalent to $r$-colorings
(see (X. Zhu, Recent developments in circular coloring of graphs, in Topics in
Discrete Mathematics Algorithms and Combinatorics Volume 26, Springer Berlin
Heidelberg (2006) 497-550)).
