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

TL;DR

The paper connects circular chromatic number and signed-graph balancing to the theory of relation algebras with up to four atoms, showing that graphs with χc < 3 are exactly spanning subgraphs of triangle-free anti-even-signable graphs and constructing a universal anti-even-signable structure. It then uses this framework to obtain a finitely bounded universal square representation for the relation algebra $56_{65}$ and proves that the corresponding network satisfaction problem lies in NP. By leveraging the generic circular triangle-free graph $\mathbb{C}_3$ and an explicit anti-even-balancing labelling of its complement, the authors establish a 3-extension property that underpins universality and links graph-theoretic characterizations to algebraic representations. The work also situates these results in a model-theoretic context, discussing homogenization and posing open questions on CSP complexity for the associated reducts and expansions, thereby bridging combinatorics, algebra, and logic.

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 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.

Figures (2)

• Figure 1: The three path configuration $\mkern 1.5mu\overline{\mkern-1.5muC_6\mkern-1.5mu}\mkern 1.5mu$
• Figure 2: Three non-isomorphic wheels over $C_5$ different from $W_5$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Truemper Truemper1982
• Example 1
• Definition 2.1
• Definition 2.2
• Lemma 2.3: Lemma 3.1 in zaslavskyDAM4
• Theorem 2.4: Proposition 3.2 in zaslavskyDAM4