Diameter critical graphs

TL;DR

This paper investigates the maximum edges in diameter-critical graphs, disproves a longstanding conjecture for diameter-2, and establishes asymptotic bounds for higher diameters, advancing extremal graph theory.

Contribution

It disproves a conjecture for diameter-2-critical graphs and provides asymptotic bounds for the maximum edges in diameter-$k$-critical graphs for all other diameters.

Findings

Disproved Caccetta and H"aggkvist's conjecture for diameter-2-critical graphs.

Established asymptotic bounds for diameter-$k$-critical graphs with $k eq 2$.

Provided an upper bound of $(rac{1}{6} + o(1))n^2$ for diameter-3-critical graphs.

Abstract

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 $(\frac{1}{6} + o(1))n^2$ for the number of edges in a diameter-3-critical graph.