E XTREMAL CASE 1 Extremal Case 1 : There exists a balanced partition - PowerPoint PPT Presentation

S OME RECENT APPLICATIONS OF S ZEMERÉDI ’ S R EGULARITY L EMMA Weihua He Department of Applied Mathematics, Guangdong University of Technology

O UTLINE 1 S ZEMERÉDI ’ S R EGULARITY L EMMA 2 L OCATING VERTICES ON H AMILTONIAN CYCLES 3 S KETCH OF THE PROOF 4 F URTHER WORKS

O UTLINE 1 S ZEMERÉDI ’ S R EGULARITY L EMMA 2 L OCATING VERTICES ON H AMILTONIAN CYCLES 3 S KETCH OF THE PROOF 4 F URTHER WORKS

R EGULAR PAIR • Density : Let G be a graph, for any two disjoint vertex sets X and Y of G . The density of the pair ( X , Y ) is the ratio d ( X , Y ) := e ( X , Y ) | X || Y | . • ǫ -regularity : Let ǫ > 0, the pair ( X , Y ) is called ǫ -regular if for every A ⊆ X and B ⊆ Y such that | A | > ǫ | X | and | B | > ǫ | Y | we have | d ( A , B ) − d ( X , Y ) | < ǫ . • Super-regularity : Let δ > 0, the pair ( X , Y ) is called ( ǫ, δ )-super-regular if it is ǫ -regular, deg Y ( x ) > δ | Y | for all x ∈ X and deg X ( y ) > δ | X | for all y ∈ Y .

P ROPERTIES OF REGULAR PAIRS L EMMA Let ( A , B ) be an ǫ -regular pair of density d and Y ⊆ B such that | Y | > ǫ | B | . Then all but at most ǫ | A | vertices in A have more than ( d − ǫ ) | Y | neighbors in Y. L EMMA (S LICING L EMMA ) ′ := max { ǫ α , 2 ǫ } . Let ( A , B ) be an ǫ -regular Let α > ǫ > 0 and ǫ ′ ⊆ A such that | A ′ | ≥ α | A | , and pair with density d. Suppose A ′ ⊆ B such that | B ′ | ≥ α | B | . Then ( A ′ , B ′ ) is an ǫ ′ -regular pair B ′ − d | < ǫ . ′ such that | d with density d

R EGULARITY LEMMA L EMMA (R EGULARITY L EMMA -D EGREE F ORM ) For every ǫ > 0 and every integer m 0 there is an M 0 = M 0 ( ǫ, m 0 ) such that if G = ( V , E ) is any graph on at least M 0 vertices and d ∈ [ 0 , 1 ] is any real number, then there is a partition of the vertex set V into l + 1 clusters V 0 , V 1 , ..., V l , and ′ = ( V , E ′ ) with the following properties: there is a subgraph G • m 0 ≤ l ≤ M 0 ; • | V 0 | ≤ ǫ | V | , and V i ( 1 ≤ i ≤ l ) are of the same size L; • deg G ′ ( v ) > deg G ( v ) − ( d + ǫ ) | V | for all v ∈ V; ′ [ V i ] = ∅ (i.e. V i is an independant set in G ′ ) for all i; • G • each pair ( V i , V j ) , 1 ≤ i < j ≤ l, is ǫ -regular, each with a density 0 or exceeding d.

R EGULARITY LEMMA

B LOW - UP L EMMA L EMMA (B LOW - UP L EMMA -B IPARTITE V ERSION ) For every δ, ∆ > 0 , there exists an ǫ = ǫ ( δ, ∆) > 0 such that the following holds. Let ( X , Y ) be an ( ǫ, δ ) -super-regular pair with | X | = | Y | = N. If a bipartite graph H with ∆( H ) ≤ ∆ can be embedded in K N , N by a function φ , then H can be embedded in ( X , Y ) . L EMMA For every δ > 0 there are ǫ BL = ǫ BL ( δ ) , n BL = n BL ( δ ) > 0 such that if ǫ ≤ ǫ BL and n ≥ n BL , G = ( A , B ) is an ( ǫ, δ ) -super-regular pair with | A | = | B | = n and x ∈ A, y ∈ B, then there is a Hamiltonian path in G starting with x and ending with y.

O UTLINE 1 S ZEMERÉDI ’ S R EGULARITY L EMMA 2 L OCATING VERTICES ON H AMILTONIAN CYCLES 3 S KETCH OF THE PROOF 4 F URTHER WORKS

L OCATING VERTICES ON H AMILTONIAN CYCLES T HEOREM (K ANEKO AND Y OSHIMOTO , 2001) Let G be a graph of order n with δ ( G ) ≥ n 2 , and let d be a positive integer such that d ≤ n 4 . Then, for any vertex subset S n with | S | ≤ 2 d , there is a Hamiltonian cycle C such that dist C ( u , v ) ≥ d for any u , v ∈ S. • The result is sharp ( | S | can not be larger) as can be seen from the graph 2 K n 2 − 1 + K 2 . When all the vertices of S are placed in one of the copies of K n 2 − 1 , then the bound becomes clear.

L OCATING VERTICES ON H AMILTONIAN CYCLES T HEOREM (S ÁRKÖZY AND S ELKOW , 2008) There are ω, n 0 > 0 such that if G is a graph with δ ( G ) ≥ n 2 on n ≥ n 0 vertices, d is an arbitrary integer with 3 ≤ d ≤ ω n 2 and S is an arbitrary subset of V ( G ) with 2 ≤ | S | = k ≤ ω n 2 , then for every sequence of integers with 3 ≤ d i ≤ d, and 1 ≤ i ≤ k − 1 , there is a Hamiltonian cycle C of G and an ordering of the vertices of S, a 1 , a 2 , ..., a k , such that the vertices of S are encountered in this order on C and we have | dist C ( a i , a i + 1 ) − d i | ≤ 1 , for all but one 1 ≤ i ≤ k − 1 . • Almost all of the distances between successive pairs of S can be specified almost exactly.

L OCATING VERTICES ON H AMILTONIAN CYCLES The two discrepancies by 1 can not be eliminated: • | dist C ( a i , a i + 1 ) − d i | ≤ 1: parity reason, e.g. G = K n 2 , S in 2 , n one side and d i is odd. • for all but one 1 ≤ i ≤ k − 1: Take two complete graphs on ′ ∪ S ′′ with S ′ ⊂ U , U and V with | U | = | V | = n 2 . Let S = S ′′ ⊂ V and | S ′′ | = | S | ′ | = | S S 2 , and add the complete ′ and V , and between S ′′ and bipartite graphs between S U .

L OCATING VERTICES ON H AMILTONIAN CYCLES T HEOREM (F AUDREE AND G OULD , 2013) Let n 1 , ..., n k − 1 be a set of k − 1 integers each at least 2 and { x 1 , ..., x k } be a fixed set of k ordered vertices in a graph G of order n. If δ ( G ) ≥ n + 2 k − 2 , then there is N = N ( k , n 1 , ..., n k − 1 ) 2 such that if n ≥ N, there is a Hamiltonian cycle C of G such that dist C ( x i , x i + 1 ) = n i for all 1 ≤ i ≤ k − 1 . • Degree condition is sharp: G = ¯ + ( n + 2 k − 3 K n − 2 k + 3 2 ( 2 k − 2 ) K 2 k − 2 ) , if 2 k vertices are all selected from one of the copies of K 2 k − 2 .

L OCATING VERTICES ON H AMILTONIAN CYCLES T HEOREM (G OULD , M AGNANT AND N OWBANDEGANI , 2017) Given an integer k ≥ 3 , let G be a graph of sufficiently large order n. Then there exists n 0 = n 0 ( k , n ) such that if n 1 , n 2 , ..., n k are a set of k positive integers with n i ≥ n 0 for all i , � n i = n, and δ ( G ) ≥ n + k 2 , then for any k distinct vertices x 1 , x 2 , ..., x k in G, there exists a Hamiltonian cycle such that the length of the path between x i to x i + 1 on the Hamiltonian cycle is n i . • Degree condition is sharp when k is even: Consider two complete graphs A and B each of order n − ( k + 1 ) . Let C be 2 the remaining k + 1 vertices. Let G = ( A + C ) ∪ ( C + B ) where the copies of vertices of C are identified. If all of the vertices x 1 , ..., x k are chosen from A and each length n i is chosen to be n k .

L OCATING PAIRS OF VERTICES ON H AMILTONIAN CYCLES C ONJECTURE (E NOMOTO ) If G is a graph of order n ≥ 3 and δ ( G ) ≥ n 2 + 1 , then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist C ( x , y ) = ⌊ n 2 ⌋ . C ONJECTURE (F AUDREE AND L I , 2012) If G is a graph of order n ≥ 3 and δ ( G ) ≥ n 2 + 1 , then for any pair of vertices x, y in G and any integer 2 ≤ k ≤ n 2 , there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k.

L OCATING PAIRS OF VERTICES ON H AMILTONIAN CYCLES C ONJECTURE (E NOMOTO ) If G is a graph of order n ≥ 3 and δ ( G ) ≥ n 2 + 1 , then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist C ( x , y ) = ⌊ n 2 ⌋ . C ONJECTURE (F AUDREE AND L I , 2012) If G is a graph of order n ≥ 3 and δ ( G ) ≥ n 2 + 1 , then for any pair of vertices x, y in G and any integer 2 ≤ k ≤ n 2 , there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k.

S HARPNESS OF THE MINIMUM DEGREE CONDITION • The degree condition is sharp. • Example 1: there is no Hamiltonian cycle such that x and y have distance n 2 . F IGURE : 2 K n 2 − 1 + K 2

S HARPNESS OF THE MINIMUM DEGREE CONDITION • The degree condition is sharp. • Example 2: x and y will be at distance n 2 in any Hamiltonian cycle of the graph. F IGURE : 2 K n 2 − 1 + K 2

L OCATING PAIRS OF VERTICES ON H AMILTONIAN CYCLES T HEOREM (F AUDREE AND L I , 2012) If p is a positive integer with 2 ≤ p ≤ n 2 and G is a graph of order n with δ ( G ) ≥ n + p 2 , then for any pair of vertices x and y in G, there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k for any 2 ≤ k ≤ p. C OROLLARY (F AUDREE AND L I , 2012) If G is a graph of order n with δ ( G ) ≥ ⌊ 3 n 4 ⌋ , then for any pair of vertices x and y of G and any positive integer 2 ≤ k ≤ ⌊ n 2 ⌋ , there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k.

L OCATING PAIRS OF VERTICES ON H AMILTONIAN CYCLES T HEOREM (F AUDREE AND L I , 2012) If p is a positive integer with 2 ≤ p ≤ n 2 and G is a graph of order n with δ ( G ) ≥ n + p 2 , then for any pair of vertices x and y in G, there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k for any 2 ≤ k ≤ p. C OROLLARY (F AUDREE AND L I , 2012) If G is a graph of order n with δ ( G ) ≥ ⌊ 3 n 4 ⌋ , then for any pair of vertices x and y of G and any positive integer 2 ≤ k ≤ ⌊ n 2 ⌋ , there is a Hamiltonian cycle C of G such that dist C ( x , y ) = k.

O UR RESULT T HEOREM (H E , L I AND S UN , 2016) There exists a positive integer n 0 such that for all n ≥ n 0 , if G is a graph of order n with δ ( G ) ≥ n 2 + 1 , then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist C ( x , y ) = ⌊ n 2 ⌋ .

O UTLINE 1 S ZEMERÉDI ’ S R EGULARITY L EMMA 2 L OCATING VERTICES ON H AMILTONIAN CYCLES 3 S KETCH OF THE PROOF 4 F URTHER WORKS