Circular coloring of signed graphs

TL;DR

This paper extends the concepts of circular coloring and chromatic number to signed graphs, establishing fundamental properties, differences from unsigned graphs, and relationships between various coloring notions.

Contribution

It introduces a new framework for circular colorings of signed graphs and analyzes their properties and differences from unsigned graph colorings.

Findings

The difference between circular and chromatic number of a signed graph is at most 1.

There exist signed graphs where this difference equals 1.

If the difference is less than 1, it is bounded away from 1 by a positive epsilon.

Abstract

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)).