[Paper Review] Cutwidth and degeneracy of graphs

Benoît Kloeckner|arXiv (Cornell University)|Jul 29, 2009

Graph theory and applications4 references3 citations

TL;DR

This paper establishes a quadratic lower bound on the cutwidth of graphs in terms of their degeneracy, using a novel inequality involving degeneracy, cutwidth, and uniform sparsity. It proves that for all graphs, cutwidth is at least $\frac{1}{4}\delta(G)^2 + \frac{1}{2}\delta(G)$, with a tighter $\frac{1}{2}\delta(G)^2$ bound for triangle-free graphs, offering a strong quantitative link between structural sparsity and layout complexity in graphs.

ABSTRACT

We prove an inequality involving the degeneracy, the cutwidth and the sparsity of graphs. It implies a quadratic lower bound on the cutwidth in terms of the degeneracy for all graphs and an improvement of it for clique-free graphs.

Motivation & Objective

• To establish a general lower bound on cutwidth in terms of degeneracy and uniform sparsity.
• To investigate how structural properties like sparsity and absence of cliques affect cutwidth complexity.
• To provide a quantitative answer to Gromov's topological question on level set complexity via graph-theoretic invariants.
• To improve existing bounds on cutwidth for specific graph classes, particularly triangle-free and $K_{k+1}$-free graphs.

Proposed method

• Introduces the concept of $(\rho,\lambda)$-uniform sparsity to control subgraph density and link it to cutwidth.
• Derives a general lower bound on cutwidth: $\mathop{\mathrm{cw}}(G) \geq \lceil\rho n\rceil\left(\delta(G) - \frac{\lceil\rho n\rceil - 1}{\lambda}\right)$.
• Applies the bound to general graphs by setting $\rho = 0$, $\lambda = 1$, yielding $\mathop{\mathrm{cw}}(G) > \frac{(\delta(G)+1)^2}{4} - 1$.
• Uses Turán’s Theorem to establish $(\rho,\lambda)$-uniform sparsity for $K_{k+1}$-free graphs, enabling tighter bounds.
• Optimizes the bound by choosing $\rho = \delta(G)/n$, leading to a quadratic lower bound for triangle-free graphs.
• Validates the bound by constructing explicit orderings for Turán graphs, showing matching upper bounds in order of magnitude.

Experimental results

Research questions

• RQ1 What is the strongest possible lower bound on cutwidth in terms of degeneracy for general graphs?
• RQ2 How does the absence of triangles or larger cliques affect the relationship between degeneracy and cutwidth?
• RQ3 Can uniform sparsity conditions be used to derive tighter quadratic bounds on cutwidth for specific graph families?
• RQ4 To what extent does the degeneracy of a graph control its cutwidth, especially in sparse or extremal graph classes?
• RQ5 How do the theoretical bounds compare with explicit constructions, such as those for Turán graphs?

Key findings

• For all simple graphs, the cutwidth satisfies $\mathop{\mathrm{cw}}(G) \geq \frac{1}{4}\delta(G)^2 + \frac{1}{2}\delta(G)$, which is sharp for complete graphs.
• For triangle-free graphs, the cutwidth is bounded below by $\frac{1}{2}\delta(G)^2$, a significant improvement over the general bound.
• For $K_{k+1}$-free graphs, the bound improves to $\mathop{\mathrm{cw}}(G) \geq \frac{k}{k-1}\frac{\delta(G)^2}{4} - \frac{k-1}{k}$, reflecting the sparsity from Turán's Theorem.
• The bound is asymptotically tight for Turán graphs $\mathrm{Tur}(n,k)$, as explicit orderings achieve cutwidth within $O(n)$ of the lower bound.
• The derived inequality shows that increasing sparsity reduces the degeneracy-to-cutwidth ratio, explaining why cutwidth grows slower than degeneracy in sparse graphs.
• The results confirm a quantitative link between topological complexity (via level set multiplicity) and combinatorial degeneracy, answering a question inspired by Gromov’s work.