Degree Form Regularity Lemma

• Degree Form Regularity Lemma is a relaxed version of Szemerédi’s Regularity Lemma that focuses on achieving ε-degularity, ensuring near-uniform vertex degrees in bipartite substructures.
• It partitions graphs into clusters with small exceptional sets while maintaining degree uniformity, even though the number of clusters grows tower-type in 1/ε.
• Extremal constructions and nested-separator techniques demonstrate that any ε-degular partition inherently requires a tower-type number of clusters, underlining the complexity in such decompositions.

The degree form regularity lemma is a relaxation of Szemerédi’s Regularity Lemma, providing a weakened but highly structured partitioning regime for graphs. The central notion—$\varepsilon$-degularity—quantifies homogeneity of vertex degrees in bipartite substructures up to small exceptional sets. Despite this relaxation, recent work demonstrates that the cost of obtaining such partitions remains inherently high, requiring a number of clusters with tower-type growth in $1/\varepsilon$ (Garbe et al., 2024).

1. Formal Definition of $\varepsilon$-Degularity

Given a finite graph $G=(V,E)$ (possibly weighted) and disjoint subsets $A,B\subset V$, the pair $(A,B)$ is said to be $\varepsilon$-degular if there exist subsets $A'\subset A$ and $B'\subset B$—called exceptional sets—with $|A'|\leq\varepsilon|A|$ and $|B'|\leq\varepsilon|B|$, such that for every $u,u'\in A\setminus A'$, $|\deg_B(u)-\deg_B(u')| \leq \varepsilon|B|$ and similarly, for $v,v'\in B\setminus B'$, $|\deg_A(v)-\deg_A(v')| \leq \varepsilon|A|$. This property ensures degree uniformity up to error $\varepsilon$, modulo exceptional vertices.

The concept extends naturally to graph partitions. A partition $V=V_1\sqcup\cdots\sqcup V_\ell$ is called an $\varepsilon$-degular partition of complexity $\ell$ if all $|V_i|$ differ by at most $1$, and for each $i$ there are at most $\varepsilon \ell$ indices $j\neq i$ for which $(V_i, V_j)$ fails to be $\varepsilon$-degular.

2. Tower-Type Lower Bound for $\varepsilon$-Degular Partitions

Garbe and Hladký establish a tower lower bound for the minimal complexity of $\varepsilon$-degular partitions. The tower function is defined recursively by $\mathrm{tower}(0)=1$, $\mathrm{tower}(x+1)=2^{\mathrm{tower}(x)}$, and, for real $x\geq1$, $\mathrm{tower}(1+x)=2^{\mathrm{tower}(x)}$.

For every $0<\varepsilon<\varepsilon_0$ and absolute constant $c>0$, there exists a graph $G=G(\varepsilon)$ such that any $\varepsilon$-degular partition of $G$ requires at least $\mathrm{tower}(c\varepsilon^{-1/3})$ clusters. The function

$$D(\varepsilon) = \sup\{\min\{\ell : G \text{ has an }\varepsilon\text{-degular partition of size } \ell\}\}$$

thus satisfies $$D(\varepsilon) \geq \mathrm{tower}(\Theta(\varepsilon^{-1/3}))$$ (Garbe et al., 2024).

3. Methodology: Extremal Constructions and Proof Sketch

The lower bound is achieved through an extremal graph construction using a "nested-separator" template (Editor's term). The construction operates in stages:

• Begin with a constant-size partition (level $0$).
• At each subsequent level $r=1,2,\ldots,s=\Theta(\varepsilon^{-1/3})$, divide each existing "blob" into $M_r$ sub-blobs, where $M_r$ grows doubly exponentially.
• Weighted edges of size $\delta=\varepsilon^{1/3}$ are assigned between oriented sub-blobs as dictated by a carefully designed separator system.

A key iterative argument shows that any $\varepsilon$-degular partition must refine nearly every level of this hierarchical partition structure. Since the last level yields $m_s\approx \mathrm{tower}(s/(4\cdot 9999))$ blobs, the partition complexity is at least this large. This methodology forces a tower-type blowup even for the more relaxed condition of degularity.

4. Comparison with Classic Regularity and Related Lower Bounds

The classic Szemerédi Regularity Lemma guarantees an $\varepsilon$-regular partition of size at most $\mathrm{tower}(O(\varepsilon^{-5}))$, with matching or nearly matching tower lower bounds $\mathrm{tower}(\Theta(\varepsilon^{-2}))$ in some variants, e.g., by Gowers, Moshkovitz, and Shapira. Even under the far weaker requirement of degularity—focusing solely on degrees rather than full bipartite density regularity—the tower-type lower bound persists but with the exponent improved to $1/3$ (Garbe et al., 2024).

The following table summarizes partition size complexity for regularity and degularity:

Partition Type	Upper Bound	Lower Bound
$\varepsilon$-regular	$\mathrm{tower}(O(\varepsilon^{-5}))$	$\mathrm{tower}(\Theta(\varepsilon^{-2}))$
$\varepsilon$-degular	(no general upper bound stated)	$\mathrm{tower}(\Theta(\varepsilon^{-1/3}))$

5. Intermediate Lemmas and Technical Properties

Several technical results undergird the lower-bound construction:

• Regular $\Rightarrow$ Degular: Every $\varepsilon$-regular pair is $2\varepsilon$-degular.
• One-Sided Density Inheritance: If $(A,B)$ is $\varepsilon$-degular, then for every $X\subseteq A$, $$|d(X,B)-d(A,B)| \leq (1+\frac{|A|}{|X|})\,\varepsilon$$.
• Separator Existence: For even $M, D$ with $M\gg 2^{D/9999}$, one can build $D$ bipartitions of $[M]$ so that each part has size $M/2$, each element appears in exactly half of the "first parts," and every pair $\{t, t'\}$ is separated in at least $(1/2-0.2)D$ bipartitions.
• Refinement Lemma: If an $\varepsilon$-degular partition nearly refines level $r-1$ in the construction, and $\delta=\varepsilon^{1/3}$ is suitably large, it also nearly refines level $r$. Iteration over all levels forces the partition to match the final, exponentially large blowup.

6. Implications and Broader Context

Despite degularity being a substantial relaxation of full regularity—focusing only on degree distributions rather than densities of pairs—towers of exponential height in $1/\varepsilon$ are still inevitably required in the worst case. This demonstrates intrinsic complexity in approximate structural decompositions of graphs, even under relaxing the precise pairwise uniformity constraints of regular partitions.

These findings position $\varepsilon$-degular partitions as a minimal, but not significantly cheaper, alternative in settings where only degree regularity is required rather than full pair regularity, and underscore the complexity barriers for algorithmic applications or theoretical analysis relying on such decompositions (Garbe et al., 2024).