Nordhaus-Gaddum inequalities for the number of cliques in a graph

TL;DR

This paper explores inequalities related to the number of cliques in a graph and its complement, extending classical Nordhaus-Gaddum bounds to multicolor and fixed-size variants, with implications for graph theory and Ramsey problems.

Contribution

It introduces new bounds for the sum and product of clique counts in multicolored graphs and fixed-size subgraph versions, expanding classical Nordhaus-Gaddum inequalities.

Findings

Derived bounds for multicolor clique sums and products.

Extended inequalities to fixed-size clique counts.

Connected results to Ramsey multiplicity and graph invariants.

Abstract

Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and product of the chromatic number of a graph and its complement. Over the years, similar inequalities have been shown for a plenitude of different graph invariants. In this paper, we consider such inequalities for the number of cliques (complete subgraphs) in a graph $G$, denoted $k(G)$. We note that some such inequalities have been well-studied, e.g., lower bounds on $k(G)+k(\overline{G})=k(G)+i(G)$, where $i(G)$ is the number of independent subsets of $G$, has been come to be known as the study of Ramsey multiplicity. We give a history of such problems. One could consider fixed sized versions of these problems as well. We also investigate multicolor versions of these problems, meaning we $r$-color the edges of $K_n$ yielding graphs $G_1,G_2,\ldots,G_r$ and give bounds on $\sum k(G_i)$ and $\prod k(G_i)$.