Packing edge disjoint cliques in graphs

TL;DR

This paper proves Győri's long-standing conjecture that graphs with slightly more edges than a Turán graph contain a proportional number of edge-disjoint r-cliques.

Contribution

The paper confirms Győri's conjecture, establishing a precise lower bound on the number of edge-disjoint r-cliques in graphs exceeding Turán graph edge counts.

Findings

Confirmed Győri's conjecture for all fixed r ≥ 3.

Established a lower bound of (2 - o(1))k/r for edge-disjoint r-cliques.

Extended understanding of clique packings in dense graphs.

Abstract

Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Gy\H{o}ri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Tur\'{a}n graph on $n$ vertices and $r - 1$ parts, then $G$ has at least $(2 - o(1))k/r$ edge disjoint $r$-cliques. We prove this conjecture.