#### Title

#### Date of Original Version

6-2014

#### Type

Working Paper

#### Abstract or Description

A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and H¨aggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs.

On the other hand, we prove that the same claim holds for all higher diameters, and is asymptotically tight, resolving the average edge-degree question in all cases except diameter-2. We also apply our techniques to prove several bounds for the original extremal question, including the correct asymptotic bound for diameter-k-critical graphs, and an upper bound of ( 1/6 +o(1))n^{2} for the number of edges in a diameter-3-critical graph.