Minimising the harmonic sum of cycle lengths

A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):={\sum }_{\mathrm{\ell }\in C(G)}1\mathrm{/}\mathrm{\ell }$ as a measure of the richness of the cycle length spectrum $C(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k,n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\le n\mathrm{/}2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\ge 2k$ satisfies $s(G)\ge {\sum }_{\mathrm{\ell }=2}^{k}1\mathrm{/}(2\mathrm{\ell })$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.