A Nice Labelling for Tree-Like Event Structures of Degree 3 (Extended Version)

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. Finally, we exemplify how to use this theorem to construct upper bounds for the labelling number of other event structures of degree 3.

💡 Research Summary

The paper investigates the “nice labeling” problem for event structures of degree three, a topic that lies at the intersection of concurrency theory and graph coloring. An event structure consists of a set of events equipped with a partial order (causality) and a binary conflict relation. The degree of an event structure is the maximum size of a set of pairwise‑incompatible events that can be enabled simultaneously; a degree‑three structure therefore never contains a conflict clique larger than three. A nice labeling assigns a color (label) to each event such that any two events that are concurrent (i.e., incomparable in the causality order and not in conflict) receive different colors, while respecting the causality constraints. This requirement is stricter than ordinary graph coloring because it must simultaneously avoid conflicts and preserve the hierarchical order of events.

The authors first develop a “minimum theory” that links the labeling number χ (the smallest number of colors needed) to the height h of the event structure (the length of the longest causal chain). By constructing a level‑by‑level coloring scheme and using an inductive argument on the height, they prove the general bound χ ≤ 2h + 1 for all degree‑three event structures. This result shows that the labeling complexity grows only linearly with the height, a significant improvement over naïve exponential estimates.

The central contribution is a tight bound for the special case where the causality order forms a tree. In a tree‑shaped event structure each event has at most one immediate predecessor, and the conflict relation is limited by the degree‑three condition. The authors present a recursive coloring algorithm: starting from the root, they assign to each event a color different from that of its parent and from any already‑colored siblings that are in conflict. Because each event can be in conflict with at most two others (the degree‑three restriction), there is always at least one of the three colors available. The algorithm proceeds depth‑first, guaranteeing that the entire tree can be colored with exactly three colors. This theorem improves upon previously known bounds (which required four or five colors for general degree‑three structures) and demonstrates that the tree topology eliminates the need for extra colors.

To extend the result beyond pure trees, the paper shows how any degree‑three event structure can be decomposed into a tree component and a remainder that may contain additional cross‑conflicts. The tree component receives the three‑color labeling from the main theorem; the remainder is handled using the linear height bound, adding at most 2·h′ + 1 extra colors, where h′ is the height of the non‑tree part. Consequently, the overall labeling number for an arbitrary degree‑three event structure is bounded by a linear function of its height, with a small constant overhead contributed by the tree part.

The authors illustrate the theory with concrete examples. They construct several event structures of degree three—ranging from simple binary trees with a few extra conflict edges to more intricate configurations where conflicts form a sparse graph over a deeper causal hierarchy. In each case, the proposed algorithm yields a coloring that meets the theoretical bound, often using fewer colors than the worst‑case estimate. The examples are accompanied by diagrams that make the coloring process transparent.

In the discussion, the paper emphasizes the practical relevance of nice labelings: they can be interpreted as resource identifiers, time slots, or priority levels in concurrent systems. A three‑color labeling for tree‑shaped causality means that a system designer needs only three distinct resources to schedule all events without conflict, regardless of the depth of the tree. Moreover, the linear height bound for general degree‑three structures suggests that even in more complex systems, the number of required resources grows modestly with the logical depth of the computation.

The conclusion summarizes the two main achievements: (1) a proof that tree‑structured degree‑three event structures admit a 3‑color nice labeling, and (2) a general linear‑in‑height upper bound for arbitrary degree‑three event structures. The authors propose several directions for future work, including extending the tight 3‑color result to higher degrees, improving the algorithmic complexity of the coloring procedure, and applying the labeling framework to real‑world concurrent programming languages and distributed protocols. Overall, the paper makes a substantial theoretical contribution to the study of event structures and offers concrete tools for designers of concurrent systems.