New results on bipartite biregular cages, block designs, and generalized polygons

In this paper, we obtain new lower and upper bounds for the problem of bipartite biregular cages. Moreover, for girth 6, we give the exact parameters of the (m, n; 6)-bipartite biregular cages when \(n\equiv -1\;\pmod m\) $n\equiv -1(modm)$ using the existence of a Steiner system \(S(2,k=m,v=1+n(m-1)+m)\) $S(2,k=m,v=1+n(m-1)+m)$ . For girth \(g=2r\) $g=2r$ and \(r=\{4,6,8\}\) $r=\{4,6,8\}$ , we use results on t-good structures given by ovoids, spreads, and sub-polygons in generalized polygons to obtain (m, n; 2r)-bipartite biregular graphs. We emphasize that, as we improve the lower bounds on the order of these graphs, we also prove that some of them are (m, n; 2r)-bipartite biregular cages. In particular, we construct relatively small bipartite biregular graphs from a special class of generalized quadrangles and hexagons. In a special case, we show that the graph obtained is a (3, 4; 8)-bipartite biregular cage on 56 vertices. Note that the order of the smallest (3, 4; 8)-graph is 39.