Minimum-Loop Realization of Degree Sequences

Given a finite sequence D of nonnegative integers, let M(D) denote its maximum element and S(D) its sum. It is known that D is realizable as a degree sequence by some graph if and only if S(D) is even, and by a loopless graph if and only if the even integer S(D) - 2M(D) ⩾ 0. Here it is shown that if the even integer 2M(D) - S(D) is positive, then one-half this integer is the minimum number of loops in graphs realizing D, and that the minimum-loop realization is unique. These results are extended to a more general loop-cost minimization problem in which loops incident at different vertices can have different costs. The possible numbers of loops, in graphs realizing D, are also determined.