Size in maximal triangle-free graphs and minimal graphs of diameter 2 Original Research Article

A triangle-free graph is maximal if the addition of any edge creates a triangle. For n ⩾ 5, we show there is an n-nodem-edge maximal triangle-free graph if and only if it is complete bipartite or 2n − 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 > no where no 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 n0 ⩾ 6 by producing a 6-node nonbipartite minimal diameter 2 graph with 8 edges.