[Paper Review] Asymptotically Optimal Vertex Ranking of Planar Graphs

Prosenjit Bose, Vida Dujmović+2 more|arXiv (Cornell University)|Jul 13, 2020

Graph Labeling and Dimension Problems27 references4 citations

TL;DR

This paper establishes asymptotically optimal bounds for ℓ-rankings in planar graphs, showing that every n-vertex planar graph admits an ℓ-ranking using O(log n / log(3) n) colors for any fixed ℓ ≥ 2, and this bound is tight for ℓ = 2. The approach leverages product structure theorems and treewidth-based decomposition, extending to bounded genus, apex-minor-free, and k-planar graphs with matching asymptotic bounds.

ABSTRACT

A (vertex) $\ell$-ranking is a colouring $φ:V(G) o\mathbb{N}$ of the vertices of a graph $G$ with integer colours so that for any path $u_0,\ldots,u_p$ of length at most $\ell$, $φ(u_0) eqφ(u_p)$ or $φ(u_0)<\max\{φ(u_0),\ldots,φ(u_p)\}$. We show that, for any fixed integer $\ell\ge 2$, every $n$-vertex planar graph has an $\ell$-ranking using $O(\log n/\log\log\log n)$ colours and this is tight even when $\ell=2$; for infinitely many values of $n$, there are $n$-vertex planar graphs, for which any 2-ranking requires $Ω(\log n/\log\log\log n)$ colours. This result also extends to bounded genus graphs. In developing this proof we obtain optimal bounds on the number of colours needed for $\ell$-ranking graphs of treewidth $t$ and graphs of simple treewidth $t$. These upper bounds are constructive and give $O(n)$-time algorithms. Additional results that come from our techniques include new sublogarithmic upper bounds on the number of colours needed for $\ell$-rankings of apex minor-free graphs and $k$-planar graphs.

Motivation & Objective

• To close the gap between known upper and lower bounds for 2-rankings of planar graphs, which prior work had left open.
• To establish tight asymptotic bounds for ℓ-rankings in planar graphs and their generalizations, including bounded genus and apex-minor-free graphs.
• To develop constructive, linear-time algorithms for computing ℓ-rankings in graphs of bounded treewidth and related classes.
• To unify and simplify the analysis of ℓ-rankings using the concept of simple treewidth, introduced via insights from David Wood.

Proposed method

• Leverages a recent product structure theorem stating that every planar graph is a subgraph of H ⊠ K₃ ⊠ P, where H has treewidth at most 3.
• Applies a key lemma showing that χℓ-vr(G₁ ⊠ G₂) ≤ χℓ-vr(G₁) · χ̄ℓ(G₂), where χ̄ℓ(G₂) is the distance-ℓ coloring number of G₂.
• Proves that for graphs of simple treewidth t, χℓ-vr(G) ∈ O(log n / log(t) n), with the bound being tight for t = 3 in planar graphs.
• Uses induction on treewidth t, with base case t = 1 (disjoint paths) requiring O(log ℓ) colors, and recursively bounds color usage on subgraphs.
• Applies the refined bound χℓ-vr(H) ∈ O(ℓ^{t−1} log^t ℓ · log n / log(t) n) to derive results for planar, bounded genus, and k-planar graphs.
• Extends results to apex-minor-free and (g,k)-planar graphs using existing product structure theorems and the refined treewidth bound.

Experimental results

Research questions

• RQ1 What is the asymptotically optimal number of colors required for a 2-ranking of an n-vertex planar graph?
• RQ2 Can the upper bound of O(log n) for ℓ-rankings of planar graphs be improved, and if so, to what extent?
• RQ3 How does the ℓ-ranking number scale with ℓ and treewidth in minor-closed graph families?
• RQ4 Can the bounds be extended to broader graph classes such as bounded genus or k-planar graphs?
• RQ5 Is there a constructive, linear-time algorithm for computing ℓ-rankings in graphs of bounded treewidth?

Key findings

• For any fixed ℓ ≥ 2, every n-vertex planar graph has an ℓ-ranking using O(log n / log(3) n) colors, and this bound is asymptotically tight for ℓ = 2.
• For infinitely many n, there exist n-vertex planar graphs requiring Ω(log n / log(3) n) colors in any 2-ranking, confirming the optimality of the upper bound.
• The bound extends to graphs of simple treewidth t: χℓ-vr(G) ∈ O(log n / log(t) n), with tightness shown for t = 3 in planar graphs.
• For graphs of treewidth t, the ℓ-ranking number is O(ℓ^t log^t ℓ · log n / log(t+1) n), which generalizes the planar result.
• The upper bounds are constructive and lead to O(n)-time algorithms for ℓ-rankings in graphs of bounded treewidth, including planar 3-trees and outerplanar graphs.
• The results extend to Euler genus ≤ g graphs (O(g log n / log(3) n)) and to apex-minor-free and (g,k)-planar graphs with bounds O(b ℓ^{c−1} log^c ℓ · log n / log(c) n) for constants b, c depending on the class.