GRAPHS WITH EVERY MATCHING CONTAINED IN A CYCLE

Let n greater-than-or-equal-to 3, 1 less-than-or-equal-to k less-than- or-equal-to n-1, and let f(n,k)=(k-1)/2+(n-k+1/2)+(k-1)2 . We prove th at if G is a graph of order n, size greater than F(n,k)=max(f(n,k),f(n ,n/2)), and minimum degree at least k, then every matching of G is con tained in a cycle of G. If k is odd and k less-than-or-equal-to (n+8)/ 6 or (n+8)/6 less-than-or-equal-to k less-than-or-equal-to n/2, and n/ 2 an odd integer, the result is the best possible. Then we give all gr aphs with minimum degree at least k and size F(n,k), having a matching which is not contained in any cycle of G.