Erdős’s exact minimisation conjecture for the harmonic cycle-length sum

Establish whether, for every pair of integers n and k with 1 ≤ k ≤ n/2, every n-vertex graph G having at least k(n−k) edges satisfies \(\sum_{\ell\in C(G)}1/\ell \geq \sum_{\ell=2}^{k}1/(2\ell)\), with equality attained by the complete bipartite graph \(K_{k,n-k}\); equivalently, determine whether \(K_{k,n-k}\) minimises the harmonic sum of cycle lengths among all such graphs.

Background

The paper studies the harmonic sum $s(G)=\sum_{\ell\in C(G)}1/\ell$, where $C(G)$ is the set of cycle lengths in a graph. Erdős conjectured that, among all n-vertex graphs with at least $k(n-k)$ edges, the complete bipartite graph $K_{k,n-k}$ minimises this quantity, whose value is $\sum_{\ell=2}^{k}1/(2\ell)$.

The main theorem proves a stronger inequality for all sufficiently large k under the edge condition $e(G)>(k-1)(n-k+1)$, and identifies $K_{k,n-k}$ as the unique equality case under the original edge threshold. Thus, the explicitly stated conjecture is established only in the sufficiently-large-k regime addressed by the paper; the universal formulation asks for the result for every admissible k and n.