  A circular $r$-coloring of a signed graph $(G, \sigma)$ is an assignment
$\phi$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G,
\sigma)$ such that for each positive edge $uv$ of $(G, \sigma)$ the distance of
$\phi(v)$ and $\phi(v)$ is at least 1 and for each negative edge $uv$ the
distance of $\phi(u)$ from the antipodal of $\phi(v)$ is at least 1. The
circular chromatic number of $(G, \sigma)$, denoted $\chi_c(G, \sigma)$, is the
infimum of $r$ such that $(G, \sigma)$ admits a circular $r$-coloring.
  This notion is recently defined by Naserasr, Wang, and Zhu who, among other
results, proved that for any signed $d$-degenerate simple graph $\hat{G}$ we
have $\chi_c(\hat{G})\leq 2d$. For $d\geq 3$, examples of signed $d$-degenerate
simple graphs of circular chromatic number $2d$ are provided. But for $d=2$
only examples of signed 2-degenerate simple graphs of circular chromatic number
close enough to $4$ are given, noting that these examples are also signed
bipartite planar graphs.
  In this work we first observe the following restatement of the 4-color
theorem: If $(G,\sigma)$ is a signed bipartite planar simple graph where
vertices of one part are all of degree 2, then $\chi_c(G,\sigma)\leq
\frac{16}{5}$. Motivated by this observation, we provide an improved upper
bound of $ 4-\dfrac{2}{\lfloor \frac{n+1}{2} \rfloor}$ for the circular
chromatic number of a signed 2-degenerate simple graph on $n$ vertices and an
improved upper bound of $ 4-\dfrac{4}{\lfloor \frac{n+2}{2} \rfloor}$ for the
circular chromatic number of a signed bipartite planar simple graph on $n$
vertices. We then show that each of the bounds is tight for any value of $n\geq
4$.
