Circular \({\boldsymbol{(4-\epsilon )}}\) -Coloring of Some Classes of Signed Graphs.

A circular -coloring of a signed graph is an assignment of points of a circle of circumference to the vertices of such that for each positive edge of the distance of from is at least 1 and for each negative edge the distance of from the antipode of is at least 1. The circular chromatic number of , denoted , is the infimum of such that admits a circular -coloring. This notion was recently defined by Naserasr, Wang, and Zhu, who, among other results, proved that for any signed -degenerate simple graph we have . For , examples of signed -degenerate simple graphs of circular chromatic number are provided. But for only examples of signed 2-degenerate simple graphs of circular chromatic number arbitrarily close 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 is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then . Motivated by this observation, we provide an improved upper bound of for the circular chromatic number of a signed 2-degenerate simple graph on vertices and an improved upper bound of for the circular chromatic number of a signed bipartite planar simple graph on vertices. We then show that each of the bounds is tight for any value of .