Concatenating bipartite graphs Paul Seymour (Princeton) joint with - PowerPoint PPT Presentation

Concatenating bipartite graphs Paul Seymour (Princeton) joint with Maria Chudnovsky, Patrick Hompe, Alex Scott and Sophie Spirkl 1 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. 2 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. True for k ≤ 2; open for k ≥ 3. 2 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. True for k ≤ 2; open for k ≥ 3. Conjecture: For every integer k ≥ 1 , if G is a bipartite digraph with n vertices in each part, and every vertex has out-degree more than n / ( k + 1 ) , then G has girth at most 2 k. 2 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. True for k ≤ 2; open for k ≥ 3. Conjecture: For every integer k ≥ 1 , if G is a bipartite digraph with n vertices in each part, and every vertex has out-degree more than n / ( k + 1 ) , then G has girth at most 2 k. Theorem (S., Spirkl, 2018) This is true for k = 1 , 2 , 3 , 4 , 6 , and all k ≥ 224 , 539 . 2 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. True for k ≤ 2; open for k ≥ 3. Conjecture: For every integer k ≥ 1 , if G is a bipartite digraph with n vertices in each part, and every vertex has out-degree more than n / ( k + 1 ) , then G has girth at most 2 k. Theorem (S., Spirkl, 2018) This is true for k = 1 , 2 , 3 , 4 , 6 , and all k ≥ 224 , 539 . True for regular digraphs. 2 / 12

Caccetta-Häggkvist conjecture, 1978 For every integer k ≥ 1 , if G is an n-vertex digraph and every vertex has out-degree at least n / k, then G has girth at most k. True for k ≤ 2; open for k ≥ 3. Conjecture: For every integer k ≥ 1 , if G is a bipartite digraph with n vertices in each part, and every vertex has out-degree more than n / ( k + 1 ) , then G has girth at most 2 k. Theorem (S., Spirkl, 2018) This is true for k = 1 , 2 , 3 , 4 , 6 , and all k ≥ 224 , 539 . True for regular digraphs. Implied by the Caccetta-Häggkvist conjecture. 2 / 12

Let G be a digraph with bipartition ( A , B ) , where every vertex in A has out-degree at least x | B | , and every vertex in B has out-degree at least y | A | . Which values of x , y guarantee girth at most four? 3 / 12

Let G be a digraph with bipartition ( A , B ) , where every vertex in A has out-degree at least x | B | , and every vertex in B has out-degree at least y | A | . Which values of x , y guarantee girth at most four? Not ( 1 / 3 , 1 / 3 ) . 3 / 12

Let G be a digraph with bipartition ( A , B ) , where every vertex in A has out-degree at least x | B | , and every vertex in B has out-degree at least y | A | . Which values of x , y guarantee girth at most four? Not ( 1 / 3 , 1 / 3 ) . 1 t Not ( 2 t + 1 , 2 t + 1 ) if t ≥ 1 is an integer. 3 / 12

Let G be a digraph with bipartition ( A , B ) , where every vertex in A has out-degree at least x | B | , and every vertex in B has out-degree at least y | A | . Which values of x , y guarantee girth at most four? Not ( 1 / 3 , 1 / 3 ) . 1 t Not ( 2 t + 1 , 2 t + 1 ) if t ≥ 1 is an integer. y GOOD ( 1 / 7 , 3 / 7 ) ( 1 / 5 , 2 / 5 ) ? ( 1 / 3 , 1 / 3 ) ? ( 2 / 5 , 1 / 5 ) BAD ( 3 / 7 , 1 / 7 ) x 3 / 12

Theorem If x , y > 0 and 2 x + y > 1 then ( x , y ) is good. 4 / 12

Theorem If x , y > 0 and 2 x + y > 1 then ( x , y ) is good. Conjecture If x , y > 0 and kx + y > 1 then G has girth at most 2 k. 4 / 12

Theorem If x , y > 0 and 2 x + y > 1 then ( x , y ) is good. Conjecture If x , y > 0 and kx + y > 1 then G has girth at most 2 k. (This implies the Caccetta-Häggkvist conjecture for the same value of k .) 4 / 12

Different question: Now G is a graph with three disjoint sets of vertices A , B , C . Every vertex in A has at least x | B | neighbours in B ; every vertex in B has at least y | C | neighbours in C . What is the maximum z such that we can guarantee some vertex in C can reach z | A | vertices in A by two-edge paths? Call this φ ( x , y ) . 5 / 12

Different question: Now G is a graph with three disjoint sets of vertices A , B , C . Every vertex in A has at least x | B | neighbours in B ; every vertex in B has at least y | C | neighbours in C . What is the maximum z such that we can guarantee some vertex in C can reach z | A | vertices in A by two-edge paths? Call this φ ( x , y ) . Theorem max ( x , y ) ≤ φ ( x , y ) . 5 / 12

Different question: Now G is a graph with three disjoint sets of vertices A , B , C . Every vertex in A has at least x | B | neighbours in B ; every vertex in B has at least y | C | neighbours in C . What is the maximum z such that we can guarantee some vertex in C can reach z | A | vertices in A by two-edge paths? Call this φ ( x , y ) . Theorem max ( x , y ) ≤ φ ( x , y ) ≤ ⌈ kx ⌉ + ⌈ ky ⌉ − 1 k for every integer k ≥ 1 . 5 / 12

Different question: Now G is a graph with three disjoint sets of vertices A , B , C . Every vertex in A has at least x | B | neighbours in B ; every vertex in B has at least y | C | neighbours in C . What is the maximum z such that we can guarantee some vertex in C can reach z | A | vertices in A by two-edge paths? Call this φ ( x , y ) . Theorem max ( x , y ) ≤ φ ( x , y ) ≤ ⌈ kx ⌉ + ⌈ ky ⌉ − 1 k for every integer k ≥ 1 . Theorem φ ( x , y ) ≤ z iff φ ( y , 1 − z ) ≤ 1 − x. 5 / 12

Different question: Now G is a graph with three disjoint sets of vertices A , B , C . Every vertex in A has at least x | B | neighbours in B ; every vertex in B has at least y | C | neighbours in C . What is the maximum z such that we can guarantee some vertex in C can reach z | A | vertices in A by two-edge paths? Call this φ ( x , y ) . Theorem max ( x , y ) ≤ φ ( x , y ) ≤ ⌈ kx ⌉ + ⌈ ky ⌉ − 1 k for every integer k ≥ 1 . Theorem φ ( x , y ) ≤ z iff φ ( y , 1 − z ) ≤ 1 − x. Theorem φ ( x , y ) = φ ( y , x ) . 5 / 12

What if also, every vertex in B has at least x | A | neighbours in A , and every vertex in C has at least y | B | neighbours in B ? Let ψ ( x , y ) be the best z . 6 / 12

What if also, every vertex in B has at least x | A | neighbours in A , and every vertex in C has at least y | B | neighbours in B ? Let ψ ( x , y ) be the best z . Theorem k + 1 < x ≤ 1 1 k then ψ ( x , x ) = 1 For every integer k > 0 , if k . 6 / 12

What if also, every vertex in B has at least x | A | neighbours in A , and every vertex in C has at least y | B | neighbours in B ? Let ψ ( x , y ) be the best z . Theorem k + 1 < x ≤ 1 1 k then ψ ( x , x ) = 1 For every integer k > 0 , if k . ψ ( x , x ) φ ( x , x ) ( 1 , 1 ) ( 1 , 1 ) ( 1 / 2 , 1 / 2 ) ( 1 / 2 , 1 / 2 ) ( 1 / 3 , 1 / 3 ) ( 1 / 3 , 1 / 3 ) ( 1 / 4 , 1 / 4 ) ( 1 / 4 , 1 / 4 ) x x 6 / 12

Does x > 1 / 3 imply φ ( x , x ) ≥ 1 / 2? 7 / 12

Does x > 1 / 3 imply φ ( x , x ) ≥ 1 / 2? Conjecture: For all integers k ≥ 1, if x , y ∈ ( 0 , 1 ] with x + ky > 1 and kx + y ≥ 1, then φ ( x , y ) ≥ 1 / k . 7 / 12

Does x > 1 / 3 imply φ ( x , x ) ≥ 1 / 2? Conjecture: For all integers k ≥ 1, if x , y ∈ ( 0 , 1 ] with x + ky > 1 and kx + y ≥ 1, then φ ( x , y ) ≥ 1 / k . Theorem If x > 1 / 3 then φ ( x , x ) > 3 / 7 . 7 / 12

Theorem If x > . 3672 then φ ( x , x ) ≥ 1 / 2 . More generally, if √ 2 x ) 2 / 2 + x > 1 then φ ( x , y ) ≥ 1 / 2 . y ( 1 + 8 / 12

Theorem If x > . 3672 then φ ( x , x ) ≥ 1 / 2 . More generally, if √ 2 x ) 2 / 2 + x > 1 then φ ( x , y ) ≥ 1 / 2 . y ( 1 + Theorem If x ≤ 1 / 3 and y < 1 / 2 and y < ( 1 − x ) 2 / ( 2 − 4 x + 6 x 2 ) , then φ ( x , y ) < 1 / 2 . 8 / 12

Theorem If x > . 3672 then φ ( x , x ) ≥ 1 / 2 . More generally, if √ 2 x ) 2 / 2 + x > 1 then φ ( x , y ) ≥ 1 / 2 . y ( 1 + Theorem If x ≤ 1 / 3 and y < 1 / 2 and y < ( 1 − x ) 2 / ( 2 − 4 x + 6 x 2 ) , then φ ( x , y ) < 1 / 2 . Theorem For all integers k ≥ 1 , if x + ky > 1 and kx + y ≥ 1 , then ψ ( x , y ) ≥ 1 / k. 8 / 12

Theorem If x > . 3672 then φ ( x , x ) ≥ 1 / 2 . More generally, if √ 2 x ) 2 / 2 + x > 1 then φ ( x , y ) ≥ 1 / 2 . y ( 1 + Theorem If x ≤ 1 / 3 and y < 1 / 2 and y < ( 1 − x ) 2 / ( 2 − 4 x + 6 x 2 ) , then φ ( x , y ) < 1 / 2 . Theorem For all integers k ≥ 1 , if x + ky > 1 and kx + y ≥ 1 , then ψ ( x , y ) ≥ 1 / k. Theorem If x + ky < 1 and kx + y < 1 , then ψ ( x , y ) < 1 k . 8 / 12

When ψ, φ ≥ 1 / 2 y y ? ( 1 / 8 , 11 / 24 ) ψ ≥ 1 / 2 ? φ ≥ 1 / 2 ( 1 / 3 , 1 / 3 ) ψ < 1 / 2 ? φ < 1 / 2 ( 8 / 17 , 1 / 6 ) ( 3 / 7 , 1 / 7 ) x x 9 / 12

When ψ, φ ≥ 2 / 3 y y y = 2 / 3 ( 1 / 5 , 3 / 5 ) ψ ≥ 2 / 3 φ ≥ 2 / 3 ( 5 / 9 , 4 / 13 ) ? ? ( 1 / 2 , 1 / 2 ) ( 1 / 2 , 1 / 2 ) ( 1 / 4 , 1 / 2 ) ( 1 / 3 , 1 / 2 ) ψ < 2 / 3 ? ( 1 / 2 , 1 / 3 ) φ < 2 / 3 ? ( 3 / 5 , 1 / 5 ) ? x x 10 / 12