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arXiv · 2310.12137

A little more about bipartite biregular cages, block designs, and generalized polygons

Gabriela Araujo-Pardo·György Kiss·Tamás Szönyi
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Abstract

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 $n\equiv -1$ $\pmod m$ using the existence of Steiner System system $S(2,k=m,v=1+n(m-1)+m)$. For girth $g=2r$ and $r=\{4,6,8\}$, 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 actually a $(3,4;8)$-bipartite biregular cage on $56$ vertices.

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Gabriela Araujo-Pardo, György Kiss, Tamás Szönyi. 2023-10-18. A little more about bipartite biregular cages, block designs, and generalized polygons. https://arxiv.org/abs/2310.12137

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