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

TL;DR

This paper investigates the circular chromatic number of signed graphs, providing new bounds for 2-degenerate and bipartite planar cases, and relates these results to the 4-color theorem.

Contribution

It introduces improved upper bounds for the circular chromatic number of certain classes of signed graphs, extending previous results and establishing tightness of these bounds.

Findings

New upper bounds for signed 2-degenerate graphs and bipartite planar graphs.

Bounds are tight for all n ≥ 4.

Connections to the 4-color theorem are discussed.

Abstract

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 $n\geq 4$.