Counting flags in triangle-free digraphs
Jan Hladký (Charles Uni, Prague & TU Munich)
Daniel Král’ (Charles Uni, Prague)
Sergey Norin (Princeton Uni)
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
I
Caccetta, Häggkvist, 1978: c = 0.3820,
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
I
Caccetta, Häggkvist, 1978: c = 0.3820,
I
Bondy, 1997: c = 0.3798,
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
I
Caccetta, Häggkvist, 1978: c = 0.3820,
I
Bondy, 1997: c = 0.3798,
I
Shen, 1998: c = 0.3543,
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
I
Caccetta, Häggkvist, 1978: c = 0.3820,
I
Bondy, 1997: c = 0.3798,
I
Shen, 1998: c = 0.3543,
I
Hamburger, Haxell, Kostochka, 2007: c = 0.3532,
digraph...no loops, no parallel edges, no counterparallel edges
minimum outdegree of a digraph D...δ + (D)
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/r contains a
(directed) cycle of length at most r .
Conjecture (Caccetta-Häggkvist, 1978)
Any digraph D on n vertices with δ + (D) ≥ n/3 contains a triangle.
Theorem
There is no triangle-free digraph D on n vertices with δ + (D) ≥ cn.
I
Caccetta, Häggkvist, 1978: c = 0.3820,
I
Bondy, 1997: c = 0.3798,
I
Shen, 1998: c = 0.3543,
I
Hamburger, Haxell, Kostochka, 2007: c = 0.3532,
I
HKN, 2009: c = 0.3465
Razborov’s theory of Flag Algebras
limits of graph...graphons
Razborov’s theory of Flag Algebras
limits of graph...graphons, graphon W : G → [0, 1]
Razborov’s theory of Flag Algebras
limits of graph...graphons, graphon W : G → [0, 1]
Razborov’s theory of Flag Algebras
limits of graph...graphons, graphon W : G → [0, 1]
Razborov’s theory of Flag Algebras
limits of graph...graphons, graphon W : G → [0, 1]
Averaging:
Razborov’s theory of Flag Algebras
limits of graph...graphons, graphon W : G → [0, 1]
Averaging:
Asymptotic version of Mantel’s Theorem
Suppose that W is a graphon with W (K2 ) > 1/2. Then
W (K3 ) > 0.
Our proof
Observation
Suppose that D is a triangle-free digraph on n vertices with
δ + (D) ≥ cn. Then for any m0 there exist m > m0 and a
triangle-free digraph D 0 on m vertices with δ + (D 0 ) ≥ cm.
Our proof
Observation
Suppose that D is a triangle-free digraph on n vertices with
δ + (D) ≥ cn. Then for any m0 there exist m > m0 and a
triangle-free digraph D 0 on m vertices with δ + (D 0 ) ≥ cm.
Main Theorem
Suppose that W is a triangle-free digraphon. Then
δ + (W ) < 0.3465.
Our proof
Observation
Suppose that D is a triangle-free digraph on n vertices with
δ + (D) ≥ cn. Then for any m0 there exist m > m0 and a
triangle-free digraph D 0 on m vertices with δ + (D 0 ) ≥ cm.
Main Theorem
Suppose that W is a triangle-free digraphon. Then
δ + (W ) < 0.3465.
Proof Take W to be a triangle-free digraphon attaining maximum
value δ + (W ) and suppose for contradiction that δ + (W ) ≥ 0.3465.
Ingredients:
I
Chudnovsky-Seymour-Sullivan, 2008: Every triangle-free
digraph with k nonedges can be made acyclic by deleting at
most k edges. improved recently by Dunkum, Hamburger, Pór
I
Cauchy-Schwarz Inequality.
I
Induction.