Monomial ideals whose all matching powers are Cohen–Macaulay

Abstract In the present article, we aim to classify monomial ideals whose all matching powers are Cohen–Macaulay. We especially focus our attention on edge ideals. The Cohen–Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analog of the famous Tutte theorem regarding graphs having a perfect matching. For chordal graphs, very well-covered graphs, and Cameron–Walker graphs, we completely solve our problem.