Decomposition of Graphs into (k,r)‐Fans and Single Edges

Let $\phi (n,H)$ be the largest integer such that, for all graphs G on n vertices, the edge set $E(G)$ can be partitioned into at most $\phi (n,H)$ parts, of which every part either is a single edge or forms a graph isomorphic to H. Pikhurko and Sousa conjectured that $\phi (n,H)=\mathrm{ex}(n,H)$ for $\chi (H)?3$ and all sufficiently large n, where $\mathrm{ex}(n,H)$ denotes the maximum number of edges of graphs on n vertices that do not contain H as a subgraph. A $(k,r)$‐fan is a graph on $(r?1)k+1$ vertices consisting of k cliques of order r that intersect in exactly one common vertex. In this article, we verify Pikhurko and Sousa's conjecture for $(k,r)$‐fans. The result also generalizes a result of Liu and Sousa.