Nordhaus–Gaddum Bounds in Graph Theory

• Nordhaus–Gaddum type bounds are extremal inequalities that relate a graph invariant for a graph and its complement, revealing duality and trade-offs in parameters such as chromatic number and spectral radius.
• These bounds have been generalized to multipart decompositions and novel invariants, broadening applications in structural, combinatorial, and spectral graph theory.
• Proof methods utilize combinatorial constructions, spectral interlacing, and optimization techniques to derive sharp bounds and characterize extremal graph structures.

A Nordhaus–Gaddum type bound is an extremal inequality relating a graph invariant evaluated on a graph and on its complement (or, more generally, on multiple disjoint decompositions). Originally formulated for the chromatic number, these inequalities have been established for a wide range of graph parameters, both classical (such as coloring, connectivity, domination) and modern (including spectral, combinatorial, and structural invariants). The underlying principle is to quantify the duality or trade-off between a parameter computed for a graph $G$ and for its complement graph $\overline{G}$, and to establish structural or algorithmic consequences that arise when the sum or product of the two is extremal.

1. Historical Context and Classical Results

The classical Nordhaus–Gaddum theorem states that for any simple graph $G$ of order $n$,

$$2\sqrt{n}\leq \chi(G)+\chi(\overline{G}) \leq n+1,\qquad n\leq\chi(G)\chi(\overline{G})\leq \left(\frac{n+1}{2}\right)^2$$

where $\chi(G)$ denotes the chromatic number. Equality in the upper bound is achieved if and only if $G$ or $\overline{G}$ is a complete or empty graph, and the structure of equality cases has been completely described. These results spurred the paper of “Nordhaus–Gaddum type” inequalities for many other invariants, a tradition now spanning several decades and resulting in both general bounds and structural characterizations of extremal cases.

2. General Framework and Extensions

The standard Nordhaus–Gaddum framework has been generalized in several important directions:

• Classical (2-part) setting: For a parameter $\beta$,

$$\beta(G) + \beta(\overline{G}), \quad \beta(G)\cdot\beta(\overline{G})$$

are bounded above or below in terms of $n$ and sometimes other graph invariants.

• Multi-part decompositions: Instead of only $G$ and $\overline{G}$, the $r$-part settings consider $r$ disjoint spanning subgraphs $G_1,\ldots,G_r$ decomposing $E(K_n)$, and paper

$$\operatorname{ngsu}_{\beta}(r;n) = \max\{\beta(G_1) + \cdots + \beta(G_r)\}$$

and

$$\operatorname{ngpu}_{\beta}(r;n) = \max\{\beta(G_1) \cdots \beta(G_r)\}$$

with analogous infima over all $r$-decompositions (Hogben et al., 2016, Caro et al., 8 May 2025).

3. Nordhaus–Gaddum Bounds Across Graph Invariants

The following table and discussion highlight representative Nordhaus–Gaddum bounds for a range of graph parameters:

Parameter	Bound Type	Formula (for simple $G$ of order $n$)	Extremal Cases/Comments
Chromatic number $\chi(G)$	sum	$2\sqrt{n} \leq \chi(G)+\chi(\overline{G}) \leq n+1$	$K_n$, $\overline{K_n}$ (Sivaraman et al., 2023)
	product	$$n \leq \chi(G)\chi(\overline{G}) \leq \left(\frac{n+1}{2}\right)^2$$
Distinguishing chromatic number $\chi_D$	sum	$\chi_D(G)+\chi_D(\overline{G}) \leq n+D(G)$	$D(G)$ is the distinguishing number (Collins et al., 2012)
Clique number $\omega(G)$	sum	$\omega(G)+\omega(\overline{G}) \leq n+1$
Maximum average degree (Mad)	$k$-decomp.	$M(k,n) < \sqrt{k} n$	Tight as $k,n\to\infty$ (Caro et al., 8 May 2025)
Location number $\beta(G)$	sum	$2 \leq \beta(G)+\beta(\overline{G}) \leq 2n-1$	$P_4$, $K_n$ and $\overline{K_n}$ (Hernando et al., 2012)
Rainbow connection number $rc(G)$	sum	$4 \leq rc(G)+rc(\overline{G}) \leq n+2$	Equality is sharp for $n\geq 4$ (upper), $n\geq 8$ (lower) (Chen et al., 2010)
Rainbow vertex-connect. $rvc(G)$	sum	$2\leq rvc(G)+rvc(\overline{G})\leq n-1$	Tight for $n\geq 5$ (Chen et al., 2011)
Total-proper conn. $tpc(G)$	sum	$6\leq tpc(G)+tpc(\overline{G})\leq n+2$	Lower bound sharp for $n\geq 4$ (Li et al., 2016)
Total-rainbow conn. $trc(G)$	sum	$trc(G)+trc(\overline{G})\leq 2n$ (for $n\geq 6$)	Examples achieve bound (Li et al., 2017)
General $G$-free chromatic $\chi_G(H)$	sum	$$\chi_G(H)+\chi_G(\overline{H})\leq \lceil n/\delta(\mathcal{G})\rceil +1$$ or $+2$	Forbidding subgraph $G$ (Rowshan, 2022)
Labeling/packing invariants	sum/product	e.g., $Y_{k-1,k,j}(G) + Y_{k-1,k,j}(\overline{G})$ bounded in terms of connectivity	(Mojdeh et al., 2016)
Signless Laplacian eigenvalues $q_2(G)$	sum	$q_2(G)+q_2(\overline{G}) \geq n-2$, $\leq 2n-5$ in many cases	Characterized extremal graphs (Huang et al., 2019)
Spectral radii $\mu(G)$	sum	$n-1\leq \mu(G)+\mu(\overline{G})\leq \sqrt{2}(n-1)$	(Elphick et al., 2016)
Eigenvalue tails $|\mu_s(G)|+|\mu_s(\overline{G})|$	sum	$\leq n/\sqrt{2(s-1)} - 1$ (for $s\geq2$ and large $n$)	Asymptotically tight (Nikiforov et al., 2014)
Complement rank $r(G) = \operatorname{rank}(A_G+I)$	sum/product	$r(G)\cdot r(\overline{G}) \geq n$, $r(G)+r(\overline{G})\geq n+1$	Sharpened for special cases (Tang, 14 Sep 2025)
Self-loop graph $G_S$ spectral radius	sum	$$n-1+\frac{2\sigma}{n}\leq \lambda_1(G_S)+\lambda_1(\overline{G_S}) \leq \ldots$$	$\sigma$ is number of looped vertices (Akbari et al., 15 May 2024)
Number of cliques $k(G)$	sum/product	$k(G)+k(\overline{G})\leq 2^n+n+1$; $k(G)\cdot k(\overline{G}) \leq (n+1)2^n$	Extremal for $K_n$, $E_n$ (Bal et al., 5 Jun 2024)
Generalized edge-connectivity $\lambda_k(G)$	sum/product	$$1\leq \lambda_k(G)+\lambda_k(\overline{G})\leq n-\lceil k/2\rceil$$	Characterized (Li et al., 2012)

The variety of parameters considered reflects both the breadth of the Nordhaus–Gaddum program and its central role in extremal combinatorics and spectral graph theory.

4. Methods and Structural Characterizations

The proofs of Nordhaus–Gaddum type statements employ a range of combinatorial, probabilistic, algebraic, and structural graph theory tools. Common approaches include:

• Edge and subgraph constructions: Extremal configurations often arise from decompositions into complete graphs, stars, or carefully balanced subgraphs (as in the construction of bounds for maximum average degree (Caro et al., 8 May 2025) or rainbow connection numbers (Chen et al., 2010)).
• Spectral interlacing and matrix methods: Many spectral results (for Laplacian, adjacency, or signless Laplacian eigenvalues) use eigenvalue interlacing, Weyl’s inequalities, or trace formulae (Nikiforov et al., 2014, Elphick et al., 2016, Huang et al., 2019).
• Degree-based partitions and algorithmic criteria: Some bounds, notably for chromatic number, use intricate vertex degree partitions and recognition algorithms specific to subclasses of extremal graphs (Collins et al., 2012).
• Compression and optimization: In clique and independent set multiplicity, “compression” (monotonicity in neighborhoods) and convex relaxation underpin the extremal threshold graph characterization (Bal et al., 5 Jun 2024).

These methods frequently yield not only the numerical bounds but also a structural description of cases where equality holds, thereby providing both extremal graphs and evidence for sharpness.

5. Applications and Significance

Nordhaus–Gaddum type inequalities serve as powerful tools in several areas:

• Structural and extremal graph theory: They connect properties of a graph and its complement, illuminating structural dualities and extremal behaviors.
• Coloring, connectivity, and domination: Many key parameters governing algorithmic complexity, robustness, or network optimization admit tight ND-type bounds, offering performance benchmarks for decomposed or adversarially structured systems.
• Spectral and algebraic combinatorics: The behavior of spectral invariants under complementation informs the design of expanders, duality in network synchronization, and the understanding of matching, independence, energy, and partitioning.
• Algorithmics and recognition: Structural characterization (e.g., of NG-graphs, threshold graphs) yields efficient algorithms for coloring, clique-finding, or domination in extremal cases (Collins et al., 2012, Sivaraman et al., 2023).
• Mathematical chemistry and network science: Invariants such as the Randić index or energy (and their bounds) translate to chemical stability, molecular branching, or centrality in complex networks (Elphick et al., 2016, Akbari et al., 15 May 2024).

6. Extensions, Open Problems, and Recent Developments

Recent work has expanded the Nordhaus–Gaddum paradigm to:

• $r$-part decompositions: Generalizing classic two-part inequalities, upper and lower bounds are established for multi-part graph decompositions, with asymptotically sharp constants depending on the parameter and the number of parts (Hogben et al., 2016, Caro et al., 8 May 2025). For example, for tree-width ($tw$),

$$\lim_{n\to\infty} \frac{\operatorname{ngsu}_{tw}(r;n)}{n} = r.$$

• Novel parameters: Recent studies include the complement rank (Tang, 14 Sep 2025), maximum average degree (Caro et al., 8 May 2025), and self-loop spectral invariants (Akbari et al., 15 May 2024), revealing new structural phenomena.
• Hereditary and forbidden subgraph characterizations: Results on hereditary NG-graphs identify graphs for which Nordhaus–Gaddum type inequalities hold for all induced subgraphs, with forbidden subgraph characterizations providing deep connections to perfect and split graph classes (Sivaraman et al., 2023).
• Hypergraph and multicolor analogs: In higher-uniformity or multicolor settings, the sum and product of invariants such as the number of lines in a hypergraph or the number of cliques in $r$-colored decompositions are tightly bounded, with extremal structures characterized through generalized compression and design techniques (Chen et al., 2014, Bal et al., 5 Jun 2024).
• Spectral open problems: Determining exactly for which $s$ the adjacency eigenvalue tail bounds are asymptotically sharp (up to additive constants) remains unresolved (Nikiforov et al., 2014). The conjectured upper bound for the sum of square roots of positive eigenvalues persists as a challenge (Elphick et al., 2016).

7. Concluding Perspectives

The program of Nordhaus–Gaddum type inequalities offers a unifying framework for quantifying the interplay between a graph and its complement—often revealing trade-offs, dualities, and surprising symmetries in both combinatorial and spectral parameters. This framework connects deeply with extremal combinatorics, spectral theory, algorithmic graph theory, and applied disciplines. Current research continues to expand both the list of parameters to which these inequalities apply and our understanding of the algebraic and structural underpinnings that control equality and sharpness phenomena across the combinatorial landscape.