On the sum of the reciprocals of cycle lengths in sparse graphs (Q1068848)

For a graph G let L(G) denote the sum of the reciprocals of cycle lengths occurring in G. In a sense, L(G) measures how rich the graph G is with respect to cycles of different lengths. For any real a let f(a) be the infimum of L(G) over all graphs G of average degree at least 2a; obviously \(f(1)=0\). In the paper it is proved that \(f((k+1)/k)=(300 k.\log k)^{-1}\) for all sufficiently large k. In particular, this implies that sparse graphs of large girth must contain many cycles of different lengths.