Indestructive deletions of edges from graphs

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Abstract

If G is a graph on p vertices with connectivity, or edge-connectivity, or minimum valency, at least k, we ask how many edges one can delete from G without reducing the appropriate quantity below k −1. When p and k are large, our answers are about 14p, 18p, and 12p, respectively; we conjecture that the correct (best possible) answer is about 12p in each case.

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This paper was written while two of us (B.B. and D.R.W.) were visiting the Department of Mathematics and Statistics at the University of Calgary, and the third (D.L.G.) was visiting the Department of Mathematics at the University of Maryland.