Abstract
A triangle-free graph is maximal if the addition of any edge creates a triangle. For n ⩾ 5, we show there is an n-node m-edge maximal triangle-free graph if and only if it is complete bipartite or 2 n − 5 ⩽ m ⩽⌊( n − 1) 2/4⌋ + 1. A diameter 2 graph is minimal if the deletion of any edge increases the diameter. We show that a triangle-free graph is maximal if and only if it is minimal of diameter 2. For n > n o where n o is a vastly huge number, Füredi showed that an n-node nonbipartite minimal diameter 2 graph has at most ⌊( n − 1) 2/4⌋ + 1 edges. We demonstrate that n 0 ⩾ 6 by producing a 6-node nonbipartite minimal diameter 2 graph with 8 edges.
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