 
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
Abstract
A graph G is ttough 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
ttough 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 t0tough 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 trianglefree 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
0
