Edge-Disjoint Cliques in Graphs with High Minimum Degree

Abstract

For a graph $G$ and a fixed integer $k \ge 3$, let $\nu_k(G)$ denote the maximum number of pairwise edge-disjoint copies of $K_k$ in $G$. For a constant $c$, let $\eta(k,c)$ be the infimum over all constants $\gamma$ such that any graph $G$ of order $n$ and minimum degree at least $cn$ has $\nu_k(G) \ge \gamma n^2(1-o_n(1))$. By Turan's theorem, $\eta(k,c)=0$ if $c \le 1-1/(k-1)$ and by Wilson's theorem, $\eta(k,c) \rightarrow 1/(k^2-k)$ as $c \rightarrow 1$. We prove that for any $1 > c > 1-1/(k-1)$, $\eta(k,c) \ge \frac{c}{2}-\frac{\big(\tbinom{k}{2}-1\big)c^{k-1}}{2\Pi_{i=1}^{k-2}((i+1)c-i)+2\big(\tbinom{k}{2}-1\big)c^{k-2}}$, while it is conjectured that $\eta(k,c) = c/(k^2-k)$ if $c \ge k/(k+1)$ and $\eta(k,c) = c/2- (k-2)/(2k-2)$ if $k/(k+1) > c > 1-1/(k-1)$. The case $k=3$ is of particular interest. In this case the bound states that for any $1 > c > 1/2$, $\eta(3,c) \ge \frac{c}{2}-\frac{c^2}{4c-1}.$ By further analyzing the case $k=3$ we obtain the improved lower bound $\eta(3,c) \ge {\frac{(12\,...