# A maximum degree theorem for diameter-2-critical graphs

Teresa Haynes; Michael Henning; Lucas Merwe; Anders Yeo

Open Mathematics (2014)

- Volume: 12, Issue: 12, page 1882-1889
- ISSN: 2391-5455

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topTeresa Haynes, et al. "A maximum degree theorem for diameter-2-critical graphs." Open Mathematics 12.12 (2014): 1882-1889. <http://eudml.org/doc/268952>.

@article{TeresaHaynes2014,

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.},

author = {Teresa Haynes, Michael Henning, Lucas Merwe, Anders Yeo},

journal = {Open Mathematics},

keywords = {Diameter critical; Diameter-2-critical; Total domination critical; diameter critical; diameter-2-critical; total domination critical},

language = {eng},

number = {12},

pages = {1882-1889},

title = {A maximum degree theorem for diameter-2-critical graphs},

url = {http://eudml.org/doc/268952},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Teresa Haynes

AU - Michael Henning

AU - Lucas Merwe

AU - Anders Yeo

TI - A maximum degree theorem for diameter-2-critical graphs

JO - Open Mathematics

PY - 2014

VL - 12

IS - 12

SP - 1882

EP - 1889

AB - 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.

LA - eng

KW - Diameter critical; Diameter-2-critical; Total domination critical; diameter critical; diameter-2-critical; total domination critical

UR - http://eudml.org/doc/268952

ER -

## References

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