How to decrease the diameter of triangle-free graphs
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Assume that G is a triangle-free graph. Let \(\) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that \(\) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of \(\) and the clique-cover number of edges of graphs. It is also shown that the maximum value of \(\) over (triangle-free) graphs of order n is \(\). The behavior of \(\) is different, its maximum value is \(\). We could not decide whether \(\) for connected (triangle-free) graphs of order n with a positive ε.
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