Arboricity Nearly Bounds Degeneracy

What the math gives to ML

The paper gives a constructive structural theorem for sparse graphs: every graph of arboricity k can be partitioned into a k-degenerate backbone and a residual graph of maximum degree at most k-1, with the more general feasibility condition d >= k and d+h >= 2k-1. This is transferable to sparse attention or graph-neural architectures as a topology constraint that separates efficiently orderable edges from bounded-degree correction edges. The useful asset is not the graph parameter itself, but the guaranteed decomposition into a low-in-degree ordered component plus a uniformly bounded residual component, which can support bounded-fan-in execution, predictable memory use, and structured pruning. The first implementation should apply the decomposition to a learned sparse token-interaction graph and compare it against unstructured top-k sparsification at equal edge count.