Circular $(4-\epsilon)$-coloring of some classes of signed graphs

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 $n\geq 4$.