Maximal edge-disjoint spanning trees in Erdős–Rényi polarity graphs

Establish that, for every prime power q, the Erdős–Rényi polarity graph ER_q attains the theoretical upper bound on the number of edge-disjoint spanning trees, namely t = q/2 when q is even and t = (q+1)/2 when q is odd, by constructing that many edge-disjoint spanning trees in ER_q for all q.

Background

The paper computes that ER_q has q^2+q+1 vertices and q(q+1)^2/2 edges, yielding a theoretical upper bound of t = q/2 for even q and t = (q+1)/2 for odd q on the number of edge-disjoint spanning trees. Using constructions based on edge-disjoint Hamiltonian paths, the authors verify that this bound is achieved up to q = 128.

Confirming that the bound is always attained for all prime powers q would strengthen the results on star-product networks (e.g., PolarStar), where maximizing edge-disjoint spanning trees in factor graphs is critical to achieving maximal or near-maximal spanning-tree packings in the product, thereby improving bandwidth for collective operations in high-performance computing.