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 Guzmán-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.

翻译：本文通过带符号图中研究的特定平衡标记，刻画了圆色数小于3的图。事实上，我们构造了一个带符号图，该图对于所有圆色数小于$3$的图的此类标记具有普适性，且与Bodirsky和Guzmán-Pro研究的泛型无圆三角形图密切相关。此外，我们的普适结构导出了关系代数$56_{65}$的一种表示。随后，我们利用该表示证明，由该关系代数描述的网络满足问题属于NP类。至此，我们完成了对至多四个原子的关系代数中普适平方表示的存在性及其对应网络满足问题复杂性的完整分类。