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Tough Ramsey graphs without short cycles AT & T Bell Labs, Murray Hill, NJ 07974, USA

Summary: Tough Ramsey graphs without short cycles
Noga Alon
AT & T Bell Labs, Murray Hill, NJ 07974, USA
and Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
A graph G is t-tough if any induced subgraph of it with x > 1 connected components is
obtained from G by deleting at least tx vertices. It is shown that for every t and g there are
t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den
Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a
conjecture of Chv´atal that there exists an absolute constant t0 so that every t0-tough graph is
pancyclic. The proof is by an explicit construction based on the tight relationship between the
spectral properties of a regular graph and its expansion properties. A similar technique provides
a simple construction of triangle-free graphs with independence number m on (m4/3
) vertices,
improving previously known explicit constructions by Erdos and by Chung, Cleve and Dagum.

Research supported in part by a United States Israel BSF Grant


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics