3-torsion in the homology of complexes of graphs of bounded degree

For δ ≥ 1 and n ≥ 1, we examine the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. δ = 1 yields the matching complex, and it is known that there is 3-torsion in degree d of the homology of this complex whenever n−4 3 ≤ d ≤ n−6 2 . We establish similar bounds for δ ≥ 2. Specifically, for δ = 2, there is 3-torsion in degree d of the homology whenever 5n−8 6 ≤ d ≤ n− 3. For δ ≥ 3, there is 3-torsion in degree d whenever (3δ−1)n−8 6 ≤ d ≤ δn−δ−6 2 and n ≥ 6δ. The situation for other pairs (d, n) remains unknown in general. To detect torsion, we construct an explicit cycle z that is easily seen to have the property that 3z is a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.