Circular Chromatic Numbers, Balanceability, Relation Algebras, and Network Satisfaction Problems

TL;DR

This paper characterizes graphs with circular chromatic number less than 3 using balanced labellings, constructs a universal signed graph, and relates it to relation algebras to analyze the complexity of associated network satisfaction problems.

Contribution

It introduces a universal signed graph for graphs with circular chromatic number less than 3 and links it to relation algebra representations to classify network satisfaction problem complexity.

Findings

Universal signed graph constructed for graphs with circular chromatic number < 3

Relation algebra representation used to analyze network satisfaction problems

Network satisfaction problem shown to be in NP for certain relation algebras

Abstract

In this paper, we characterize graphs with circular chromatic number less than 3 in terms of certain balancing labellings studied in the context of signed graphs. In fact, we construct a signed graph which is universal for all such labellings of graphs with circular chromatic number less than $3$, and is closely related to the generic circular triangle-free graph studied by Bodirsky and Guzm\'an-Pro. Moreover, our universal structure gives rise to a representation of the relation algebra $56_{65}$. We then use this representation to show that the network satisfaction problem described by this relation algebra belongs to NP. This concludes the full classification of the existence of a universal square representation, as well as the complexity of the corresponding network satisfaction problem, for relation algebras with at most four atoms.