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 q is an integer, we construct a doubly–infinite series of (q, q)–bipartite graphs which are biregular of degrees q and q and of girth 8. These graphs have the greatest number of edges among all known (n,m)–bipartite graphs with the same asymptotics of lognm, n → ∞. For s = 1/3, our graphs provide an explicit counterexample to a conjecture of Erdős which states that an (n,m)–bipartite graph with m = O(n) and girth at least 8 has O(n) edges. This conjecture was recently disproved by de Caen and Székely [2], who established the existence of a family of such graphs having n edges. Our graphs have n edges, and so come closer to the best known upper bound of O(n).