Bipartite Biregular Cages and Block Designs

Abstract

A bipartite biregular (n, m; g)-graph G is a bipartite graph of even girth g having the degree set {n, m} and satisfying the additional property that the vertices in the same partite set have the same degree. An (n, m; g)bipartite biregular cage is a bipartite biregular (n, m; g)-graph of minimum order. In their 2019 paper, Filipovski, Ramos-Rivera and Jajcay present lower bounds on the orders of bipartite biregular (n, m; g)-graphs, and call the graphs that attain these bounds bipartite biregular Moore cages.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 S(2, k, v)-Steiner systems yields the existence of bipartite biregular (k, v−1 k−1 ; 6)-Moore cages. Moreover, in the special case of Steiner triple systems (i.e., in the case k = 3), we completely solve the problem of the existence of (3, m; 6)-bipartite biregular cages for all integers m ≥ 4.Considering girths higher than 6 and prime powers s, we relate the existence of generalized polygons (quadrangles, hexagons and octagons) with the existence of (n + 1, n 2 + 1; 8), (n + 1, n 3 + 1; 12), and (n + 1, n 2 +

Graph Spanners in the Message-Passing Model

“…For bipartite biregular graphs of larger girth, the optimal size of the graph is unknown. However, a simple counting argument known as the Moore bound gives a lower bound on the number of vertices of a bipartite biregular graph of prescribed bidegree and girth [Hoo02,FRJ19,APRRJ19], which shows limitations of the above technique for proving communication lower bounds for multiplicative spanners of larger distortion. More specifically, let g = 2k + 2 be the girth and let the two degrees be {d, sd} (as we require one side of the bipartite graph to be of size Θ(n/s) in our proof technique).…”