A condition for a graph to contain k-matching Original Research Article

Let k, l, n be nonnegative integers such that 1⩽k⩽n/2, and let G be a graph of order n with the minimum vertex-degree δ(G)⩾l. We prove that if the size e(G) of G verifies e(G)>F(n,k,l)=max{f(n,k,l),f(n,k,k−1)}, where f(n,k,l)=2k−l−12+1(l(n−2k+l+1) then G contains kK2. Moreover, if e(G)=F(n,k,l) and G contains no kK2 then l⩽k−1 and G=K2k−2p−1∗Kp∗K̄n−2k+p+1, where p∈{l,k−1}. We conjecture a similar statement for forests with at most k edges.