Seminar of Graph Theory - Róbert Jajcay (5.12.2019)

Thursday 5.12.2019 at 9:50, Lecture room M/213

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Róbert Jajcay:
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 our talk we 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,n2+1;8), (n+1,n3+1;12), and (n+1,n2+1;16)-bipartite biregular Moore cages, respectively. Using this connection, we derive improved upper bounds for the orders of bipartite biregular cages of girths 8, 12 and 14.

This is joint work with Alejandra Ramos Rivera, Slobodan Filipovski and Gabriela Araujo Pardo.

 

 

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