#### Title

A Maximum Degree Theorem for Diameter-2-Critical Graphs

#### Document Type

Article

#### Publication Date

1-1-2014

#### Description

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.

#### Citation Information

Haynes, Teresa W.;
Henning, Michael A.;
van der Merwe, Lucas C.;
and
Yeo, Anders.
2014.
A Maximum Degree Theorem for Diameter-2-Critical Graphs.
*Central European Journal of Mathematics*.
Vol.12(12).
1882-1889.
https://doi.org/10.2478/s11533-014-0449-3
ISSN: 1895-1074