A Nice Labelling for Tree-Like Event Structures of Degree 3

TL;DR

This paper proves that event structures of degree 3 with a tree causality order can be nicely labeled with 3 colors, and establishes bounds on labeling numbers based on structure height.

Contribution

It introduces a minimal theory showing the labeling number of degree 3 event structures is linearly bounded by height, and proves a specific 3-color labeling for tree-like structures.

Findings

Event structures of degree 3 with tree causality have a 3-color labelling.

Labelling number is linearly bounded by the height of the structure.

Method to derive upper bounds for other degree 3 event structures.

Abstract

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.