How to decrease the diameter of triangle-free graphs

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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Received: October 12, 1997

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Erdős, P., Gyárfás, A. & Ruszinkó, M. How to decrease the diameter of triangle-free graphs. Combinatorica 18, 493–501 (1998). https://doi.org/10.1007/s004930050035

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  • AMS Subject Classification (1991) Classes:  05C12, 05C35