In this paper, we study the decompositions of a graph G into edge-disjoint subgraphs all of which belong to a specified class of graphs $\mathcal{H}$. Let $\alpha (G;\mathcal{H})$ denote the minimum value of the total sum of the sizes of subgraphs in $\mathcal{H}$ into which G can be decomposed, taken over all such decompositions of G. Let $\alpha (n;\mathcal{H})$ denote the maximum value of $\alpha (G;\mathcal{H})$ over all graphs G with n vertices.In this paper, we settle a conjecture of Katona and Tarjan by showing \[ \alpha (n;\mathcal{K}) = \left\lfloor {n^2 /2} \right\rfloor ,\] where $\mathcal{K}$ denotes the set of all complete graphs. Moreover, we show that the complete bipartite graph G on $\lfloor n/2 \rfloor $ and $\lceil n/2 \rceil $ vertices is the only graph with $\alpha (G;\mathcal{K}) = \alpha (n;\mathcal{K})$.