Graphs of prescribed girth and bi-degree (Q1898720)

A bipartite graph \(\Gamma(V_1\cup V_2, E)\) is said to have bi- degree \(r\), \(s\) if every vertex from \(V_1\) has degree \(r\) and every vertex from \(V_2\) has degree \(s\). \(\Gamma\) is called an \((r, s, t)\)- graph if, additionally, the girth of \(\Gamma\) (i.e. the length of a shortest cycle of \(\Gamma\)) is \(2t\). For \(t> 3\), very few examples of \((r, s, t)\)-graphs are known. In this paper, we give a recursive construction of \((r, s, t)\)-graphs for all \(r, s, t\geq 2\), as well as an algebraic construction of such graphs for all \(r, s\geq t\geq 3\).