On the difference between clique partition and clique covering numbers of graphs

For a graph $G$, let $\text{cpn}(G)$ and $\text{ccn}(G)$ denote the minimum numbers of cliques whose edge sets partition and cover $E(G)$, respectively, and put $f(n)={\max }_{\mathrm{\mid }V(G)\mathrm{\mid }=n}(\text{cpn}(G)-\text{ccn}(G))\mathrm{.}$ In 1983, Erd\H{o}s, Faudree, and Ordman asked whether there is a sequence of graphs $G_n$ such that $\mathrm{\mid }V(G_n)\mathrm{\mid }=n$ and $\text{cpn}(G_n)-\text{ccn}(G_n)=n^2\mathrm{/}4+O(n)$. The question appears as Problem 66 in Chung’s survey \cite{ChungProblems} and is also listed on the UCSD Erd\H{o}s Problems website. Caccetta, Erd\H{o}s, Ordman, and Pullman proved that $f(n)=n^2\mathrm{/}4-o(n^2)$. We prove that $f(n)=\lfloor \frac{n^2}{4}\rfloor -\mathrm{\Theta }(n^{4\mathrm{/}3}),$ and hence answer the question in the negative.