A maximum degree theorem for diameter-2-critical graphs

Teresa W. Haynes, Michael A. Henning, Lucas C. van der Merwe, Anders Yeo

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235-240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81-98] proved the conjecture for n > n 0 where n 0 is a tower of 2's of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

Original languageEnglish
JournalCentral European Journal of Mathematics
Volume12
Issue number12
Pages (from-to)1882-1889
ISSN1895-1074
DOIs
Publication statusPublished - 2014
Externally publishedYes

Keywords

  • Diameter critical
  • Diameter-2-critical
  • Total domination critical

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