# A high girth graph construction and a lower bound for hitting set size for combinatorial rectangles

Chandran, Sunil L (1999) A high girth graph construction and a lower bound for hitting set size for combinatorial rectangles. In: Proceedings 19th FST & TCS 1999: Nineteenth Conference on the Foundations of Software Technology and Theoretical Computer Science, 13-15 Dec. 1999, Chennai, India, pp. 283-290.

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## Abstract

We give the following two results. First, we give a deterministic algorithm which constructs a graph of girth $log_k(n)+O(1)$ and minimum degree k-1, taking number of nodes n and the number of edges e=[nk/2] as input. The graphs constructed by our algorithm are expanders of sub-linear sized subsets, that is subsets of size at most $n^\delta$, where $\delta$<¼. Although methods which construct high girth graphs are known, the proof of our construction uses only very simple counting arguments in comparison. Also our algorithm works for all values of n or k. We also give a lower bound of m/8&epsiv; for the size of hitting sets for combinatorial rectangles of volume &epsiv;. This result is an improvement of the previously known lower bound, namely Ω(m+1/&epsiv;+log(d)). The known upper bound for the size of the hitting set is m poly(log(d)/&epsiv;) (N. Linial et al., 1997)

Item Type: Conference Paper Copyright of this article belongs to Springer-Verlag computational complexity:graph theory;set theory;theorem proving Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 25 Jul 2007 27 Aug 2008 12:42 http://eprints.iisc.ernet.in/id/eprint/10273