On the Turán number of Kλ1∪Kλ2

Given a graph H, the Turán number ex(n,H) of H is the maximum number of edges of an n-vertex simple graph containing no H as a subgraph. Let Kλ1∪Kλ2 be the disjoint union of Kλ1 and Kλ2. For λ1=λ2=ρ+1≥3, Chen, Lu and Yuan determined the value ex(n,Kλ1∪Kλ2) completely. In this paper, we consider ex(n,Kλ1∪Kλ2) for λ1>λ2. We first give a lower bound on ex(n,Kλ1∪Kλ2), and then determine the value ex(n,Kλ1∪Kλ2) for λ1>λ2, λ2=2,3 and n≥λ1+λ2.