London’s spanning-tree extremal conjecture for C4-free graphs

Determine whether, for every prime power q and n=q^2+q+1, the maximum number of spanning trees among n-vertex C4-free graphs equals n^{(n-3)/2}, with equality attained precisely by the orthogonal polarity graphs.

Background

For every prime power q, the Erdős–Rényi orthogonal polarity graph ER_q is C4-free and has n^{(n-3)/2} spanning trees, providing a general lower bound for the extremal quantity st(n,C4). London conjectured that this lower bound is always sharp and that the only extremal graphs are orthogonal polarity graphs.

The paper proves the conjecture for every prime power q outside the exceptional set {7,9,11,13}. For the four exceptional values, the uniform theorem establishes the desired bound for graphs with at most m_q=\frac12q(q+1)^2 edges, but does not settle graphs with more than m_q edges; consequently, the conjecture remains unresolved in those cases.