Sparsity-Parameterised Dynamic Edge Colouring

TL;DR

The paper addresses dynamic edge colouring in graphs by parameterising the palette size with the graph's arboricity $α$. It introduces a deterministic fully dynamic algorithm that maintains a proper $Δ+O(α)$ edge-colouring with polylogarithmic amortized update time, adapting to the current $Δ$ and $α$, and complements this with a static result achieving $Δ(uv)+2α$ colours in $O(m\log n)$ time along with a local colouring property. The approach hinges on hierarchical partitions (H-partitions) and a level-based data structure, including multiple palettes per vertex to efficiently recolour edges and maintain validity under updates. The work also demonstrates near-term improvements for special graph classes such as forests and planar graphs, and discusses independence with concurrent work, worst-case vs amortized guarantees, and open questions for further reductions in palette size and update times.

Abstract

We study the edge-colouring problem, and give efficient algorithms where the number of colours is parameterised by the graph's arboricity, $α$. In a dynamic graph, subject to insertions and deletions, we give a deterministic algorithm that updates a proper $Δ+ O(α)$ edge~colouring in $\operatorname{poly}(\log n)$ amortized time. Our algorithm is fully adaptive to the current value of the maximum degree and arboricity. In this fully-dynamic setting, the state-of-the-art edge-colouring algorithms are either a randomised algorithm using $(1 + \varepsilon)Δ$ colours in $\operatorname{poly}(\log n, ε^{-1})$ time per update, or the naive greedy algorithm which is a deterministic $2Δ-1$ edge colouring with $\log(Δ)$ update time. Compared to the $(1+\varepsilon)Δ$ algorithm, our algorithm is deterministic and asymptotically faster, and when $α$ is sufficiently small compared to $Δ$, it even uses fewer colours. In particular, ours is the first $Δ+O(1)$ edge-colouring algorithm for dynamic forests, and dynamic planar graphs, with polylogarithmic update time. Additionally, in the static setting, we show that we can find a proper edge colouring with $Δ+ 2α$ colours in $O(m\log n)$ time. Moreover, the colouring returned by our algorithm has the following local property: every edge $uv$ is coloured with a colour in $\{1, \max\{deg(u), deg(v)\} + 2α\}$. The time bound matches that of the greedy algorithm that computes a $2Δ-1$ colouring of the graph's edges, and improves the number of colours when $α$ is sufficiently small compared to $Δ$.

Figures (5)

• Figure 1: Hierarchical partition of the vertex set of a graph $G$.
• Figure 2: Hierarchical partition. In the static setting, the out degree of a vertex is bounded by $d$.
• Figure 3: We may need to recolour at most $l(u)$ edges
• Figure 4: Hierarchical partition with two levels. The in degree of a vertex $v$ is still only bounded by $\Delta$, when the bound on the out degree depends on its group $g(v)$.
• Figure 5: When the degree of a vertex $v$ decrease, we may need to recolour at most $l(v)$ edges

Theorems & Definitions (20)

• Theorem 1: Palettes
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 9
• Lemma 10: Maximum level