On a relation between bipartite biregular cages, block designs and generalized polygons

Abstract

A bipartite biregular $\left(m,n;g\right)$-graph $\Gamma$ is a bipartite graph of even girth $g$ having the degree set $\left\{m,n\right\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An $\left(m,n;g\right)$-bipartite biregular cage is a bipartite biregular $\left(m,n;g\right)$-graph of minimum order. In their 2019 paper, Filipovski, Ramos-Rivera, and Jajcay present lower bounds on the orders of bipartite biregular $\left(m,n;g\right)$-graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well-known classical results relating the existence of $k$-regular Moore graphs of even girths $g=6,8$, and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an $S\left(2,k,v\right)$-Steiner system yields the existence of a bipartite biregular $\left(k,\frac{v-1}{k-1};6\right)$-cage, and, vice versa, the existence of a bipartite biregular $\left(k,n;6\right)$-cage whose order is equal to one of our lower bounds yields the existence of an $S\left(2,k,1+n\left(k-1\right)\right)$-Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of $\left(3,n;6\right)$-bipartite biregular cages for all integers $n\ge 4$. Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of $\left(n+1,n^2+1;8\right)$-, $\left(n^2+1,n^3+1;8\right)$-, $\left(n,n+2;8\right)$-, $\left(n+1,n^3+1;12\right)$- and $\left(n+1,n^2+1;16\right)$-bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.