# A High Girth Graph Construction

Chandran, Sunil L (2003) A High Girth Graph Construction. In: SIAM Journal on Discrete Mathematics, 16 (3). pp. 366-370.

 PDF high.pdf Restricted to Registered users only Download (106Kb) | Request a copy

## Abstract

We give a deterministic algorithm that constructs a graph of girth $log_k(n) + O(1)$ and minimum degree k − 1, taking number of nodes n and number of edges e =[nk/2] (where $k< \frac{n}{3}$)as input. The degree of each node is guaranteed to be k − 1, k, or k + 1, where k is the average degree. Although constructions that achieve higher values of girth—up to $\frac{4}{33}$ $log_{k-1}$ (n)—with the same number of edges are known, the proof of our construction uses only very simple counting arguments in comparison. Our method is very simple and perhaps the most intuitive: We start with an initially empty graph and keep introducing edges one by one, connecting vertices which are at large distances in the current graph. In comparison with the Erd¨os–Sachs proof, ours is slightly simpler while the value it achieves is slightly lower. Also, our algorithm works for all values of n and $k<\frac{n}{3}$, unlike most of the earlier constructions.

Item Type: Journal Article Copyright of Society for Industrial and Applied Mathematics. Girth;Algorithm Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 02 Jun 2006 19 Sep 2010 04:28 http://eprints.iisc.ernet.in/id/eprint/7306