Cohen-Macaulay Simplicial Complexes of Degree k

For a positive integer k a class of simplicial complexes, to be denoted by CM(k), is introduced. This class generalizes Cohen-Macaulay simplicial complexes. In analogy with the Cohen-Macaulay complexes, we give some homological and combinatorial properties of CM(k) complexes. It is shown that the complex ? is CM(k) if and only if I?? , the Stanley-Reisner ideal of the Alexander dual of ?, has a k-resolution, i.e. Bi. j(I?? ) = 0 unless j = ik+q, where q is the degree of I?? . As a main result, we characterize all bipartite graphs whose independence complexes are CM(k) and show that an unmixed bipartite graph is CM(k) if and only if it is pure k-shellable. Our result improves a result due to Herzog and Hibi and also a result due to Villarreal.