New Constructions of Bipartite Graphs on m, n Vertices with Many Edges and Without Small Cycles

For arbitrary odd prime power q and s ∈ (0, 1] such that qs is an integer, we construct a doubly-infinite series of (q5, q3 + s)-bipartite graphs which are biregular of degrees qs and q2 and of girth 8. These graphs have the greatest number of edges among all known (n, m)-bipartite graphs with the same amsymptotics of lognm, n → ∞. For s = 13, our graphs provide an explicit counterexample to a conjecture of Erdős which states that an (n, m)-bipartite graph with m = O(n23) and girth at least 8 has O(n) edges. This conjecture was recently disproved by de Caen and Székely, who established the existence of a family of such graphs having n1 + 157 + o(1) edges. Our graphs have n1 + 115 edges, and so come closer to the best known upper bound of O(n1 + 19).