Nordhaus-Gaddum-type theorems for maximum average degree

Yair Caro, Zsolt Tuza

Published 2025-05-08Version 1

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 $\chi(G_1) + \chi(G_2)$ and $\chi(G_1) \chi(G_2)$ 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.