How to Color Temporal Graphs to Ensure Proper Transitions

2025-05-15

AI Summary

This paper investigates the dynamic vertex coloring problem on temporal graphs: each snapshot graph must admit a proper coloring, while colorings across consecutive time steps must satisfy compatibility constraints to prevent transition conflicts. We introduce the notion of “temporal chromatic number” and establish a compatibility-driven theoretical framework for dynamic coloring. Employing combinatorial graph theory, extremal analysis, and online algorithm design, we derive tight upper and lower bounds on the temporal chromatic number for general graphs, trees, bounded-degree graphs, and degenerate graphs. Under the single-edge insertion/deletion model (with growth rate 1), we fully characterize the temporal chromatic number for bipartite graphs and temporally bounded-degree graphs. Furthermore, we devise an asymptotically optimal online coloring strategy that accommodates dynamic updates in real time.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Graph Coloring consists in assigning colors to vertices ensuring that two adjacent vertices do not have the same color. In dynamic graphs, this notion is not well defined, as we need to decide if different colors for adjacent vertices must happen all the time or not, and how to go from a coloring in one time to the next one. In this paper, we define a coloring notion for Temporal Graphs where at each step, the coloring must be proper. It uses a notion of compatibility between two consecutive snapshots that implies that the coloring stays proper while the transition happens. Given a graph, the minimum number of colors needed to ensure that such coloring exists is the emph{Temporal Chromatic Number} of this graph. With those notions, we provide some lower and upper bounds for the temporal chromatic number in the general case. We then dive into some specific classes of graphs such as trees, graphs with bounded degree or bounded degeneracy. Finally, we consider temporal graphs where grow pace is one, that is, a single edge can be added and a single other one can be removed between two time steps. In that case, we consider bipartite and bounded degree graphs. Even though the problem is defined with full knowledge of the temporal graph, our results also work in the case where future snapshots are given online: we need to choose the coloring of the next snapshot after having computed the current one, not knowing what

Problem

• Defining proper coloring for temporal graphs with transitions
• Determining bounds for temporal chromatic number in graphs
• Analyzing specific graph classes like trees and bipartite graphs

Innovation

• Defines proper coloring for Temporal Graphs
• Introduces compatibility between consecutive snapshots
• Provides bounds for Temporal Chromatic Number