Graphs of Prescribed Girth and Bi-Degree

Abstract

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.