A strengthening of the Murty-Simon Conjecture on diameter 2 critical - PowerPoint PPT Presentation

A strengthening of the Murty-Simon Conjecture on diameter 2 critical graphs Antoine Dailly 1 , Florent Foucaud 2 , Adriana Hansberg 3 1 Universit´ e Lyon 1, LIRIS, Lyon, France 2 LIMOS, Universit´ e Clermont Auvergne, Aubi` ere, France 3 Instituto de Matem´ aticas, UNAM, Mexico 1/20 ICGT, July 11, 2018

Diameter 2 critical graphs 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. Diameter 2 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. Diameter d critical graphs A graph is diameter d critical (or D d C) if: 1. It has diameter d ; 2. Deleting any edge increases the diameter. Diameter 2 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. Diameter d critical graphs A graph is diameter d critical (or D d C) if: 1. It has diameter d ; 2. Deleting any edge increases the diameter. Diameter 2 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. Diameter d critical graphs A graph is diameter d critical (or D d C) if: 1. It has diameter d ; 2. Deleting any edge increases the diameter. Diameter 2 2/20

Diameter 2 critical graphs Diameter The diameter of a graph is the highest distance between two vertices. Diameter d critical graphs A graph is diameter d critical (or D d C) if: 1. It has diameter d ; 2. Deleting any edge increases the diameter. Diameter 2 critical 2/20

Diameter 2 critical graphs Several well-known graphs: 3/20

Diameter 2 critical graphs Several well-known graphs: • • • • • • Complete bipartite graphs Clebsch Graph Chv` atal Graph 3/20

Diameter 2 critical graphs Several well-known graphs: • • • • • • Complete bipartite graphs Clebsch Graph Chv` atal Graph ... and many others! 3/20

The Murty-Simon Conjecture 4/20

The Murty-Simon Conjecture Theorem (Mantel, 1907) A triangle-free graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n 4/20

The Murty-Simon Conjecture Theorem (Mantel, 1907) A triangle-free graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n Diameter 2 triangle-free graphs ⇔ D2C triangle-free graphs 4/20

The Murty-Simon Conjecture Theorem (Mantel, 1907) A triangle-free graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n Diameter 2 triangle-free graphs ⇔ D2C triangle-free graphs Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n 4/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) ◮ m < 0 . 2532 n 2 (Fan, 1987) 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) ◮ m < 0 . 2532 n 2 (Fan, 1987) The conjecture holds for: ◮ n ≤ 24, n = 26 (Fan, 1987) 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) ◮ m < 0 . 2532 n 2 (Fan, 1987) The conjecture holds for: ◮ n ≤ 24, n = 26 (Fan, 1987) ◮ n ≥ 2 2 ... 2 size 10 14 (F¨ uredi, 1992) 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) ◮ m < 0 . 2532 n 2 (Fan, 1987) The conjecture holds for: ◮ n ≤ 24, n = 26 (Fan, 1987) ◮ n ≥ 2 2 ... 2 size 10 14 (F¨ uredi, 1992) ◮ ∆ ≥ 0 . 6756 n (Jabalameli et al. , 2016) 5/20

The Murty-Simon Conjecture: history Conjecture (Murty, Simon, Ore, Plesn´ ık, 1970s) A D2C graph of order n and size m verifies m ≤ ⌊ n 2 4 ⌋ . The extremal graph is K ⌊ n 2 ⌉ . 2 ⌋ , ⌈ n = 0 . 375( n 2 − n ) (Plesn´ ◮ m < 3 n ( n − 1) ık, 1975) 8 √ 12 n 2 < 0 . 27 n 2 (Cacceta, H¨ ◮ m < 1+ 5 aggkvist, 1979) ◮ m < 0 . 2532 n 2 (Fan, 1987) The conjecture holds for: ◮ n ≤ 24, n = 26 (Fan, 1987) ◮ n ≥ 2 2 ... 2 size 10 14 (F¨ uredi, 1992) ◮ ∆ ≥ 0 . 6756 n (Jabalameli et al. , 2016) ◮ With a dominating edge (Hanson and Wang, 2003, Haynes et al. , 2011, Wang 2012) 5/20

The Murty-Simon Conjecture: a linear strengthening? 6/20

The Murty-Simon Conjecture: a linear strengthening? Claim (F¨ uredi, 1992) A non-bipartite D2C graph of order n ≥ n 0 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1 ≈ ⌊ n 2 4 − n 2 ⌋ . The extremal graph is obtained by 4 subdividing an edge of K ⌊ n − 1 2 ⌉ . 2 ⌋ , ⌈ n − 1 • • • • • • 6/20

The Murty-Simon Conjecture: a linear strengthening? Claim (F¨ uredi, 1992) A non-bipartite D2C graph of order n ≥ n 0 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1 ≈ ⌊ n 2 4 − n 2 ⌋ . The extremal graph is obtained by 4 subdividing an edge of K ⌊ n − 1 2 ⌉ . 2 ⌋ , ⌈ n − 1 I n − 3 2 I I • • • • • • Theorem (Balbuena et al. , 2015) A triangle-free non-bipartite D2C graph of order n and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1. The extremal graphs are some inflations 4 of C 5 . 6/20

Strengthening the Murty-Simon Conjecture 7/20

Strengthening the Murty-Simon Conjecture Conjecture: linear strengthening (Balbuena et al. , 2015) A non-bipartite D2C graph of order n > 6 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1. If n ≥ 10, the extremal graphs are some 4 inflations of C 5 . 7/20

Strengthening the Murty-Simon Conjecture Conjecture: linear strengthening (Balbuena et al. , 2015) A non-bipartite D2C graph of order n > 6 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1. If n ≥ 10, the extremal graphs are some 4 inflations of C 5 . Conjecture: constant strengthening (D., Foucaud, Hansberg, 2018) Let c be a positive integer, then there is a rank n 0 such that any non-bipartite D2C graph of order n ≥ n 0 and size m verifies m < ⌊ n 2 4 ⌋ − c . 7/20

Strengthening the Murty-Simon Conjecture Conjecture: linear strengthening (Balbuena et al. , 2015) A non-bipartite D2C graph of order n > 6 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1. If n ≥ 10, the extremal graphs are some 4 inflations of C 5 . Conjecture: constant strengthening (D., Foucaud, Hansberg, 2018) Let c be a positive integer, then there is a rank n 0 such that any non-bipartite D2C graph of order n ≥ n 0 and size m verifies m < ⌊ n 2 4 ⌋ − c . Asymptotical since small graphs may not verify the strengthened conjectures. n = 6 H 5 = m = 8 7/20

Strengthening the Murty-Simon Conjecture Conjecture: linear strengthening (Balbuena et al. , 2015) A non-bipartite D2C graph of order n > 6 and size m verifies m ≤ ⌊ ( n − 1) 2 ⌋ + 1. If n ≥ 10, the extremal graphs are some 4 inflations of C 5 . Conjecture: constant strengthening (D., Foucaud, Hansberg, 2018) Let c be a positive integer, then there is a rank n 0 such that any non-bipartite D2C graph of order n ≥ n 0 and size m verifies m < ⌊ n 2 4 ⌋ − c . Asymptotical since small graphs may not verify the strengthened conjectures. m > ⌊ ( n − 1) 2 ⌋ + 1 = 7 n = 6 4 m = 8 ⇒ H 5 = 7/20