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Given a finite simple graph one can associate the edge ideal. In this paperwe prove that a graded Betti number of the edge ideal does not vanish if theoriginal graph contains a set of complete bipartite subgraphs with someconditions. Also we give a combinatorial description for the projectivedimension of the edge ideals of unmixed bipartite graphs.

Non-vanishing of Betti numbers of edge ideals and complete bipartite subgraphs