Cutwidth and degeneracy of graphs

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.

💡 Research Summary

The paper establishes a quantitative link between two fundamental graph parameters – the cut‑width (cw) and the degeneracy (δ) – by introducing a third parameter that measures uniform sparsity. The motivation comes from a question posed by M. Gromov: given a continuous map from a high‑complexity topological space (or, in the combinatorial setting, a graph) to the real line, how large must the multiplicity of the level sets be? In graph theory the maximal multiplicity of a continuous map G→ℝ coincides with the cut‑width, defined as the smallest possible maximum number of edges crossing a cut induced by a linear ordering of the vertices.

The paper first recalls the standard definition of cut‑width: for an ordering O=(x₁<…<x_n) of V(G), cw(G,O)=max_i |{uv∈E(G) : u≤x_i<v}|, and cw(G)=min_O cw(G,O). It then defines the degeneracy δ(G) as the largest integer k for which the k‑core of G is non‑empty; equivalently, δ(G) is the maximum value of the minimal degree that can be forced by repeatedly deleting vertices of degree <k. Degeneracy is well known to bound the chromatic and list‑chromatic numbers.

To obtain a non‑trivial lower bound on cw(G) in terms of δ(G), the author introduces the notion of (ρ,λ)‑uniform sparsity. A graph on n vertices is λ‑sparse if it has at most n(n−1)/(2λ) edges. It is (ρ,λ)‑uniformly sparse if every induced subgraph with at least ρn vertices is λ‑sparse. This definition allows one to control the edge density of large subgraphs while ignoring tiny dense pieces (e.g., a single edge).

Theorem 2.1 (main result). Let G be a simple graph on n vertices that is (ρ,λ)‑uniformly sparse. Then

1. cw(G) > ⌈ρn⌉·(δ(G) − (⌈ρn⌉−1)/λ).  (1)

2. If 2 n ρ ≤ δ(G)λ − 1, then
  cw(G) > (δ(G)λ + 1)²/(4λ) − (δ(G)λ + 1)/(2λ).  (2)

The proof proceeds by considering the δ‑core G′ of G, which has minimum degree δ(G) and satisfies cw(G) ≥ cw(G′). For an optimal ordering O of G′, let n_i be the number of edges crossing the cut after the i‑th vertex. Uniform sparsity guarantees that for i>ρn the induced subgraph on the first i vertices contains at most i(i−1)/(2λ) edges, while the minimum degree condition forces the sum of degrees in that subgraph to be at least i·δ(G). Combining these yields a lower bound on n_i, which after evaluating at i=⌈ρn⌉ gives (1), and at i≈(δ(G)λ+1)/2 (provided the inequality on ρ holds) yields (2).

From this general theorem the author derives several concrete corollaries:

• Corollary 2.2 (general graphs). Since every graph is trivially (0,1)‑uniformly sparse, (2) with λ=1 gives
   cw(G) > δ(G)²/4 + δ(G)/2.
  This shows that cut‑width grows at least quadratically with degeneracy for any graph. The bound is tight for complete graphs K_n, where cw(K_n)=⌊n²/4⌋ and δ(K_n)=n−1.

• Corollary 2.3 (triangle‑free graphs). Turán’s theorem implies that any triangle‑free graph on n vertices is 2n−1/n‑sparse, and consequently (ρ,λ)‑uniformly sparse with λ≈2. Applying (1) with ρ≈δ(G)/n yields
   cw(G) > ½ δ(G)².
  This improves the general bound by a factor of two for graphs without K₃.

• Corollary 2.4 (K_{k+1}‑free graphs). Using the general Turán bound for K_{k+1}‑free graphs, one obtains a family‑dependent sparsity parameter λ=k/(k−1). Substituting into (1) gives
   cw(G) > (k/(k−1))·δ(G)²/4 − (k−1)·δ(G)/k.
  For the extremal Turán graph T_{ur}(n,k) (the most balanced complete k‑partite graph) this lower bound is asymptotically sharp; an explicit vertex ordering shows that cw(T_{ur}(n,k))≈(k−1)/k·n²/4+O(n).

The paper also includes a short discussion of circular cut‑width, proving that for trees the circular and linear cut‑width coincide, and uses this fact to give a concise proof of a known result on trees.

Overall, the work provides the first systematic quadratic lower bound on cut‑width in terms of degeneracy, and demonstrates how uniform sparsity can be leveraged to strengthen the bound under forbidden‑subgraph conditions. This not only answers Gromov’s original question in the combinatorial setting—high‑degeneracy graphs necessarily have high‑multiplicity level sets under any map to ℝ—but also opens a pathway for further investigations linking structural graph parameters (such as expansion, tree‑width, or eigenvalue gaps) to cut‑width.