Graphs of Prescribed Girth and Bi-Degree

We say that a bipartite graph Γ(V1 ∪ V2, E) has bi-degree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)-graph if, additionally, the girth of Γ is 2t. For t > 3, very few examples of (r, s, t)-graphs were previously known. In this paper we give a recursive construction of (r, s, t)-graphs for all r, s, t ≥ 2, as well as an algebraic construction of such graphs for all r, s ≥ t ≥ 3.