[HTML][HTML] Extremal properties of regular and affine generalized m-gons as tactical configurations

VA Ustimenko, AJ Woldar - European Journal of Combinatorics, 2003 - Elsevier
VA Ustimenko, AJ Woldar
European Journal of Combinatorics, 2003Elsevier
The purpose of this paper is to derive bounds on the sizes of tactical configurations of large
girth which provide analogues to the well-known bounds on the sizes of graphs of large
girth. Let exα (v, g) denote the greatest number of edges in a tactical configuration of order v,
bidegree a, aα and girth at least g. We establish the upper bound ex α (v, g)= O (v 1+ 1 τ),
where τ= 1 4 (α+ 1) g− 1 for g≡ 0 (mod4) and τ= 1 4 (α+ 1) g+ 1 2 (α− 3) for g≡ 2 (mod4).
We further demonstrate this bound to be sharp for the regular and affine generalized m-gons …
The purpose of this paper is to derive bounds on the sizes of tactical configurations of large girth which provide analogues to the well-known bounds on the sizes of graphs of large girth. Let exα(v,g) denote the greatest number of edges in a tactical configuration of order v, bidegree a, aα and girth at least g. We establish the upper bound ex α(v,g)=O(v 1+1 τ ) , where τ= 1 4 (α+1)g−1 for g≡0(mod4) and τ= 1 4 (α+1)g+ 1 2 (α−3) for g≡2(mod4). We further demonstrate this bound to be sharp for the regular and affine generalized m-gons but not for the nonregular generalized m-gons. Finally, we derive lower bounds on exα(v,g) via explicit group theoretic constructions.
Elsevier
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