Many cliques with few edges

TL;DR

This paper characterizes the structure of graphs with a fixed number of edges and maximum degree that maximize the number of cliques of size at least two, extending previous vertex-focused results to edge constraints.

Contribution

It introduces a new extremal graph characterization for maximizing large cliques under edge and degree constraints, generalizing prior vertex-based theorems.

Findings

Graphs with fixed edges and max degree maximize large cliques as disjoint unions of cliques plus a colex graph.

The extremal structure is a union of equal-sized cliques and a residual colex graph.

The result extends known vertex-based clique maximization to edge-based constraints.

Abstract

Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1$ together with the colex graph using the remainder of the edges.