Abstract
We give the following two results. First, we give a deterministic algorithm which constructs a graph of girth logk(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 δ, where δ < 1/4. Though 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/8ε for the size of hitting sets for combinatorial rectangles of volume ε. This result is an improvement of the previously known lower bound, namely. (m + 1 /ε + log(d)). The known upper bound for the size of the hitting set is m poly(log(d)/ε). [LLSZ].
Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, 560012. This work is based on the author’s masters thesis.
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Chandran, L.S. (1999). A High Girth Graph Construction and a Lower Bound for Hitting Set Size for Combinatorial Rectangles. In: Rangan, C.P., Raman, V., Ramanujam, R. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1999. Lecture Notes in Computer Science, vol 1738. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46691-6_22
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