Abstract:In this paper, we characterize finite graphs with circular chromatic number less than 3 in terms of the existence of certain signings ($\mathbb Z_2$-labellings studied in the context of signed graphs). In fact, we construct a signed graph which is universal for all such signings -- called anti-triangle-signings in this paper -- of finite $\overline{K_3}$-free graphs, 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.
From: Matěj Konečný [view email]
[v1]
Sun, 7 Dec 2025 15:09:07 UTC (18 KB)
[v2]
Wed, 2 Sep 2026 15:09:15 UTC (19 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。