Minimum cuts, girth and a spectral threshold
Information Processing Letters
Journal of Combinatorial Theory Series B
From High Girth Graphs to Hard Instances
CP '08 Proceedings of the 14th international conference on Principles and Practice of Constraint Programming
On balanced CSPs with high treewidth
AAAI'07 Proceedings of the 22nd national conference on Artificial intelligence - Volume 1
On spanners of geometric graphs
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
European Journal of Combinatorics
Edge-Partitioning Regular Graphs for Ring Traffic Grooming with a Priori Placement of the ADMs
SIAM Journal on Discrete Mathematics
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We give a deterministic algorithm that constructs a graph of girth logk(n) + O(1) and minimum degree k-1, taking number of nodes n and number of edges $e = {\left \lfloor nk / 2 \right \rfloor }$ (where $k k-1, k, or k+1, where k is the average degree. Although constructions that achieve higher values of girth---up to $\frac {4}{3} \log_{k-1}{(n)}$---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 Erdös--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