How to Color Temporal Graphs to Ensure Proper Transitions

TL;DR

The paper proposes a compatibility-based framework for coloring temporal graphs, introducing the temporal chromatic number $\\chi^t$ and the grow-pace concept to model edge changes over time. It derives fundamental upper bounds via 3-smashed unions and links temporal coloring to static coloring through constructive reductions, while also exploring online feasibility. The work analyzes several graph classes (trees, degeneracy, bounded degree, bipartite) and a grow-pace-1 regime, providing tight or near-tight bounds (e.g., $\\chi^t \le 8$ for bipartite snapshots, $5d \le \\chi^t \le 12d$ for $d$-degenerate snapshots, and $\\chi^t \le \\Delta+2$ for grow-pace-1 with maximum degree $\\Delta$). It also presents online algorithms and highlights open questions about gaps between lower and upper bounds, particularly for trees and larger degree regimes. Overall, the paper establishes a versatile framework for robust dynamic coloring with clear connections to classical coloring and practical implications for online and streaming settings.

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

Figures (7)

• Figure 1: Temporal coloring $4$ consecutive snapshots $G_{1},G_2,G_{3},G_{4}$. The number $1,2,3,4$ inside every vertex is the color of that vertex in the corresponding snapshots.
• Figure 3: A temporal graph with all snapshots being bipartite and temporal chromatic number 8.
• Figure 4: Two paths: $H_1$ (dotted blue line), $H_2$ (dashed red line), such that $H_1\cup H_2$ is a clique.
• Figure 5: Three consecutive snapshots: $G_1$ (dotted blue line), $G_2$ (dashed red line), $G_3$ (solid black line), which are paths. The 3-smashed graph $G_1 \cup G_2 \cup G_3$ is the complete graph on six vertices.
• Figure 6: Three consecutive paths: a part of $G_2$ (dotted lines), a part of $G_3$ (solid lines), a part of $G_4$ (dashed lines). A node $v$ labeled $a$ means $c_2(v)=a$; a node $v$ labeled $a \rightarrow b$ means that $c_2(v) = a$ and $\{u \mid uv \in E_2 \cup E_3\}=[6]\setminus \{a,b\}$.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 7
• Theorem 8
• Lemma 9
• Theorem 10
• Theorem 11
• Lemma 12