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- W2951925166 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 Haggkvist (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." @default.
- W2951925166 created "2019-06-27" @default.
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- W2951925166 date "2014-06-25" @default.
- W2951925166 modified "2023-10-16" @default.
- W2951925166 title "Diameter critical graphs" @default.
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- W2951925166 doi "https://doi.org/10.48550/arxiv.1406.6736" @default.
- W2951925166 hasPublicationYear "2014" @default.
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