Graphs of Prescribed Girth and Bi-Degree

We say that a bipartite graph Γ( V 1 ∪ V 2, E) has bi-degree r, s if every vertex from V 1 has degree r and every vertex from V 2 has degree s. Γ is called an ( r, s, t)-graph if, additionally, the girth of Γ is 2 t. 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.