Nordhaus-Gaddum-type theorems for maximum average degree

A $k$-decomposition $(G_1,\dots,G_k)$ of a graph $G$ is a partition of its edge set into $k$ spanning subgraphs $G_1,\dots,G_k$. The classical theorem of Nordhaus and Gaddum bounds $χ(G_1) + χ(G_2)$ and $χ(G_1) χ(G_2)$ over all 2-decompositions of $K_n$. For a graph parameter $p$, let $p(k,G) = \max { \sum_{i=1}^k p(Gi) }$, taken over all $k$-decompositions of graph $G$. In this paper we consider $M(k,K_n) = M(k,n) = \max { \sum_{i=1}^k \mathrm{Mad}(G_i) }$, taken over all $k$-decompositions of the complete graph $K_n$, where $\mathrm{Mad}(G)$ denotes the maximum average degree of $G$, $\mathrm{Mad}(G) = \max { 2e(H)/|H| : H \subseteq G } = \max {d(H) : H \subseteq G }$. Among the many results obtained in this paper we mention the following selected ones. (1) $M(k, n) < \sqrt{k} n$, and $\lim_{k\to\infty} ( \liminf_{n\to\infty} \frac{M(k,n)}{\sqrt{k},n} ) = 1$. (2) Exact determination of $M(2,n)$. (3) Exact determination of $M(k,n)$ when $k = \binom{n}{2} - t$, $0 \leq t\leq (n-1)^2/3$. Applications of these bounds to other parameters considered before in the literature are given.

💡 Research Summary

The paper studies a Nordhaus‑Gaddum‑type extremal problem for the maximum average degree (Mad) of graphs. For a graph G, Mad(G)=max_{H⊆G} 2e(H)/|H|, i.e., the largest average degree over all subgraphs. Given a complete graph K_n, a k‑decomposition is a partition of its edge set into k spanning subgraphs G_1,…,G_k. The authors define

M(k,n)=max_{all k‑decompositions of K_n} Σ_{i=1}^k Mad(G_i),

and also a “list” variant

M_L(k,N)=max_{lists of k graphs with total edge count N} Σ Mad(G_i).

The main contributions are:

1. General Upper Bound – Using a continuous relaxation and Lagrange multipliers, they prove M(k,n) < √k·n for all admissible k,n (Lemma 14, Corollary 17). This matches the trivial lower bound up to a factor of √k.

2. Exact Value for k=2 – Theorem 18 gives a closed formula for M(2,n): M(2,n)=⌊n⌋+⌊n/2⌋−1. The extremal decomposition consists of one subgraph that is a complete graph on roughly n/2+1 vertices and a second subgraph that contains the remaining edges arranged to maximise its own Mad.

3. Near‑Complete Decompositions – When k = C(n,2) – t with 0 ≤ t ≤ (n−1)²/3, Theorem 21 (Section 5.2) determines M(k,n) exactly. The result shows that deleting t edges from the full edge set reduces the total Mad by essentially ⌈t/(n−1)⌉, reflecting the fact that each removed edge can lower the average degree of at most one subgraph by at most 2/(n−1).

4. Complete Determination of the List Function – Theorem 20 (Section 5.1) expresses M_L(k,N) in closed form. Writing N = k·p² + q·p + r with 0 ≤ q < k and 0 ≤ r < p, they obtain

   • M_L = k·p – k + q if r ≤ (p−1)/2,
   • M_L = k·p – k + q + 1 – 2(p−r)/(p+1) if r ≥ (p−1)/2. This piecewise linear formula captures the optimal distribution of edges among the k graphs when only the total edge count is prescribed.

5. Sufficient Condition for Equality M(k,n)=M_L(k, C(n,2)) – Theorem 21 provides a combinatorial condition (essentially the existence of a suitable Steiner system or block design) under which the list optimum can be realized by an actual decomposition of K_n. When this condition holds, the upper bound from the list model becomes tight for the original problem.

6. Construction Techniques – The authors develop a toolbox based on combinatorial designs:

   • Steiner systems S(2, p, n) and related finite geometries to distribute edges evenly.
   • Wilson’s decomposition theorem for partitioning complete graphs into prescribed subgraphs.
   • (p, p+1)-block designs and substitution methods that replace each point of a small design by a larger structure. These constructions yield many families of (k,n) where M(k,n)=M_L(k, C(n,2)). In particular, for fixed k they prove (Theorem 34) that (1−o_k(1))√k·n − c_k < M(k,n) < √k·n, establishing asymptotic tightness and confirming that lim_{k→∞} liminf_{n→∞} M(k,n)/(√k·n) = 1.

7. Applications to Other Graph Parameters – Since Mad(G)+1 dominates several classic invariants (clique number ω, chromatic number χ, choice/chromatic number ch, Szekeres‑Wilf number col), the bounds on M(k,n) immediately translate into Nordhaus‑Gaddum‑type results for these parameters. Theorem 40 formalizes this: whenever M(k,n)=M_L(k, C(n,2)), the extremal sums for the mentioned parameters equal either M(k,n) or ⌊M(k,n)⌋+k.

8. Broader Impact and Open Problems – The paper highlights that the list‑variant approach is of independent interest and can be adapted to other parameters (e.g., connectivity, arboricity). The concluding section lists several open directions, such as extending the method to non‑spanning subgraphs, exploring tighter lower bounds for specific k, and investigating analogous problems for hypergraphs.

Overall, the work fills a notable gap in the Nordhaus‑Gaddum literature by treating the maximum average degree, introduces a versatile list‑based framework, and leverages deep combinatorial design theory to obtain exact formulas, asymptotic estimates, and a suite of applications. It provides both a comprehensive theoretical foundation and concrete constructions that will likely inspire further research in extremal graph theory and design theory.