Strictly Completing Partial Latin Squares Jaromy Kuhl Department of - - PowerPoint PPT Presentation

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Strictly Completing Partial Latin Squares Jaromy Kuhl Department of Mathematics and Statistics University of West Florida May 2011 Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 1 / 25 Contents Introduction 1 Jaromy

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Strictly Completing Partial Latin Squares

Jaromy Kuhl

Department of Mathematics and Statistics University of West Florida

May 2011

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 1 / 25

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Contents

1

Introduction

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 2 / 25

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SLIDE 3

Contents

1

Introduction

2

Strictly Completing

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 2 / 25

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SLIDE 4

Contents

1

Introduction

2

Strictly Completing

3

Main Result

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 2 / 25

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SLIDE 5

Contents

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 2 / 25

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SLIDE 6

Contents

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 2 / 25

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Introduction

Current Section

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 3 / 25

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SLIDE 8

Introduction

partial latin squares

Definition 1 A partial latin square of order n is an n × n array of n symbols so that each symbol appears at most once in each row and column.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 4 / 25

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SLIDE 9

Introduction

partial latin squares

Definition 1 A partial latin square of order n is an n × n array of n symbols so that each symbol appears at most once in each row and column. 1 3 5 4 3 1 3 3 1 3 4 1

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 4 / 25

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SLIDE 10

Introduction

partial latin squares

Definition 1 A partial latin square of order n is an n × n array of n symbols so that each symbol appears at most once in each row and column. 1 3 5 4 3 1 3 3 1 3 4 1 1 2 3 4 5 2 4 1 5 3 5 1 2 3 4 4 3 5 1 2 3 5 4 2 1

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 4 / 25

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Introduction

completing partial latin squares

Theorem 1 (Smetaniuk, 1981) Every partial latin square of order n with at most n − 1 entries is completable.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 5 / 25

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Introduction

avoiding partial Latin squares

Definition 2 A partial Latin square P of order n is called avoidable if there is a Latin square L of order n such that on every set of n symbols L contains no part of P.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 6 / 25

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Introduction

avoiding partial Latin squares

Definition 2 A partial Latin square P of order n is called avoidable if there is a Latin square L of order n such that on every set of n symbols L contains no part of P. 1 1 2 3 4 1 5 2 3 4

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 6 / 25

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SLIDE 14

Introduction

avoiding partial Latin squares

Definition 2 A partial Latin square P of order n is called avoidable if there is a Latin square L of order n such that on every set of n symbols L contains no part of P. 1 1 2 3 4 1 5 2 3 4 2 3 5 1 4 1 2 4 5 3 5 1 3 4 2 3 4 1 2 5 4 5 2 3 1

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 6 / 25

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Introduction

avoiding partial latin squares

Theorem 1 Every partial Latin square of order k ≥ 4 is avoidable.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 7 / 25

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Introduction

avoiding partial latin squares

Theorem 1 Every partial Latin square of order k ≥ 4 is avoidable. 1 2

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 7 / 25

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SLIDE 17

Introduction

avoiding partial latin squares

Theorem 1 Every partial Latin square of order k ≥ 4 is avoidable. 1 2 1 2 3 3 1 2 1

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 7 / 25

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Strictly Completing

Current Section

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 8 / 25

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Strictly Completing

strictly completing partial latins squares

Definition 3 Let P and Q be partial latin squares of order n that avoid each other. We say that P is strictly completable with respect to Q if P can be completed to a Latin square L and L avoids Q.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 9 / 25

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SLIDE 20

Strictly Completing

strictly completing partial latins squares

Definition 3 Let P and Q be partial latin squares of order n that avoid each other. We say that P is strictly completable with respect to Q if P can be completed to a Latin square L and L avoids Q. Conjecture 1 Let P and Q be partial latin squares of order n > 3 that avoid each

  • ther. If P contains at most n − 2 entries, then P can is strictly

completable with respect to Q.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 9 / 25

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SLIDE 21

Strictly Completing

strictly completing partial latins squares

Definition 3 Let P and Q be partial latin squares of order n that avoid each other. We say that P is strictly completable with respect to Q if P can be completed to a Latin square L and L avoids Q. Conjecture 1 Let P and Q be partial latin squares of order n > 3 that avoid each

  • ther. If P contains at most n − 2 entries, then P can is strictly

completable with respect to Q. 1 2 3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 9 / 25

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SLIDE 22

Strictly Completing

strictly completing partial latins squares

Definition 3 Let P and Q be partial latin squares of order n that avoid each other. We say that P is strictly completable with respect to Q if P can be completed to a Latin square L and L avoids Q. Conjecture 1 Let P and Q be partial latin squares of order n > 3 that avoid each

  • ther. If P contains at most n − 2 entries, then P can is strictly

completable with respect to Q. 1 2 3 4

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 9 / 25

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Main Result

Current Section

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 10 / 25

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Main Result

Theorem 2 Let k = 4t for t ≥ 9 be a positive integer.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 11 / 25

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Main Result

Theorem 2 Let k = 4t for t ≥ 9 be a positive integer. Let P and Q be partial latin squares of order k such that P and Q avoid each other.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 11 / 25

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Main Result

Theorem 2 Let k = 4t for t ≥ 9 be a positive integer. Let P and Q be partial latin squares of order k such that P and Q avoid each other. If P contains at most t − 1 entries, then P can be strictly completed with respect to Q.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 11 / 25

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SLIDE 27

Main Result

preliminary results

Lemma 3 Let P and Q be partial latin squares of order 4 that avoid each other and let P contain at most one entry. Then P can be strictly completed with respect to Q provided

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 12 / 25

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Main Result

preliminary results

Lemma 3 Let P and Q be partial latin squares of order 4 that avoid each other and let P contain at most one entry. Then P can be strictly completed with respect to Q provided 1. Q contains at most 3 symbols, or

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 12 / 25

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Main Result

preliminary results

Lemma 3 Let P and Q be partial latin squares of order 4 that avoid each other and let P contain at most one entry. Then P can be strictly completed with respect to Q provided 1. Q contains at most 3 symbols, or 2. Q contains 4 symbols of which at least one appears only

  • nce.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 12 / 25

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Main Result

preliminary results

Definition 4 Let X be a partial array of symbols of order 4.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 13 / 25

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Main Result

preliminary results

Definition 4 Let X be a partial array of symbols of order 4. A 4-tuple of symbols is called bad in X if each symbol in the 4-tuple appears at least twice in X.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 13 / 25

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Main Result

preliminary results

Definition 4 Let X be a partial array of symbols of order 4. A 4-tuple of symbols is called bad in X if each symbol in the 4-tuple appears at least twice in X. Lemma 4 Let x be a symbol appearing in X. There are at most 20 bad 4-tuples in X containing x.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 13 / 25

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Main Result

theorem of Daykin and Häggkvist

Theorem 5 Let 0 ≤ d < k and let H be an r-partite r-uniform hypergraph with minimum degree δ(H) and |V(H)| = rk. If δ(H) > r − 1 r

  • kr−1 − (k − d)r−1

, then H has more than d independent edges.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 14 / 25

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Proof outline for main result

Current Section

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 15 / 25

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Proof outline for main result

We may suppose that

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 16 / 25

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Proof outline for main result

We may suppose that 1. the symbols appearing in P come from the set {1, 2, . . . , t},

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 16 / 25

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Proof outline for main result

We may suppose that 1. the symbols appearing in P come from the set {1, 2, . . . , t}, 2. no two entries appear in the same 4 × 4 subsquare of P, and

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 16 / 25

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Proof outline for main result

We may suppose that 1. the symbols appearing in P come from the set {1, 2, . . . , t}, 2. no two entries appear in the same 4 × 4 subsquare of P, and 3. no symbol appears in two 4 × 4 subsquares of P sharing the same rows or columns.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 16 / 25

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Proof outline for main result

We may suppose that 1. the symbols appearing in P come from the set {1, 2, . . . , t}, 2. no two entries appear in the same 4 × 4 subsquare of P, and 3. no symbol appears in two 4 × 4 subsquares of P sharing the same rows or columns. 1 1 2 3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 16 / 25

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Proof outline for main result

Let T be a partial latin square of order t on the symbol set {X1, . . . , Xt} such that

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 17 / 25

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Proof outline for main result

Let T be a partial latin square of order t on the symbol set {X1, . . . , Xt} such that cell (j, l) contains Xi if and only if i appears in the corresponding 4 × 4 subsquare of P.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 17 / 25

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Proof outline for main result

Let T be a partial latin square of order t on the symbol set {X1, . . . , Xt} such that cell (j, l) contains Xi if and only if i appears in the corresponding 4 × 4 subsquare of P. 1 1 2 3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 17 / 25

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Proof outline for main result

Let T be a partial latin square of order t on the symbol set {X1, . . . , Xt} such that cell (j, l) contains Xi if and only if i appears in the corresponding 4 × 4 subsquare of P. 1 1 2 3 X1 X1 X2 X3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 17 / 25

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Proof outline for main result

Let T be a partial latin square of order t on the symbol set {X1, . . . , Xt} such that cell (j, l) contains Xi if and only if i appears in the corresponding 4 × 4 subsquare of P. 1 1 2 3 X1 X1 X2 X3 Since T contains at most t − 1 entries, T can be completed.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 17 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that 1. |Si| = 4 for each i ∈ [t],

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that 1. |Si| = 4 for each i ∈ [t], 2. i ∈ Si for each i ∈ [t],

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that 1. |Si| = 4 for each i ∈ [t], 2. i ∈ Si for each i ∈ [t], 3. there are latin squares of order 4 on Si avoiding the 4 × 4 subsquares in Q corresponding to each Xi, and

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that 1. |Si| = 4 for each i ∈ [t], 2. i ∈ Si for each i ∈ [t], 3. there are latin squares of order 4 on Si avoiding the 4 × 4 subsquares in Q corresponding to each Xi, and 4. there are latin squares of order 4 on Si completing the 4 × 4 subsquares in P corresponding to each Xi.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

We wish to find a partition S1, S2, . . . , St of [4t] such that 1. |Si| = 4 for each i ∈ [t], 2. i ∈ Si for each i ∈ [t], 3. there are latin squares of order 4 on Si avoiding the 4 × 4 subsquares in Q corresponding to each Xi, and 4. there are latin squares of order 4 on Si completing the 4 × 4 subsquares in P corresponding to each Xi.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 18 / 25

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Proof outline for main result

1 1 2 3 X1 X1 X2 X3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 19 / 25

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Proof outline for main result

1 1 2 3 X1 X1 X2 X3 X1 X1 X1 X2 X1 X1 X3

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 19 / 25

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Proof outline for main result

1 1 2 3 X1 X1 X2 X3 X1 X1 X1 X2 X1 X1 X3 If such a partition can be found, then P is strictly completable with respect to Q.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 19 / 25

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SLIDE 54

Proof outline for main result

Let H be a 4-partite, 4-uniform hypergraph with vertex set {A, B, C, D} such that

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 20 / 25

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Proof outline for main result

Let H be a 4-partite, 4-uniform hypergraph with vertex set {A, B, C, D} such that 1. A = {(Xi, i) : i ∈ [t]}, 2. B = {t + 1, . . . , 2t}, 3. C = {2t + 1, . . . , 3t}, and 4. D = {3t + 1, . . . , 4t}.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 20 / 25

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Proof outline for main result

Let H be a 4-partite, 4-uniform hypergraph with vertex set {A, B, C, D} such that 1. A = {(Xi, i) : i ∈ [t]}, 2. B = {t + 1, . . . , 2t}, 3. C = {2t + 1, . . . , 3t}, and 4. D = {3t + 1, . . . , 4t}. The edge ((Xi, i), b, c, d) is included in H if and only if {i, b, c, d} is not a bad 4-tuple for each 4 × 4 subsquare of Q corresponding to Xi in T.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 20 / 25

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Proof outline for main result

dH((Xi, i)) ≥ t3 − 20t

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 21 / 25

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SLIDE 58

Proof outline for main result

dH((Xi, i)) ≥ t3 − 20t δ(H) ≥ t3 − 20t

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 21 / 25

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SLIDE 59

Proof outline for main result

dH((Xi, i)) ≥ t3 − 20t δ(H) ≥ t3 − 20t According to the theorem of Daykin and Häggkvist; H has t independent edges provided δ(H) > 3

4(t3 − 1).

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 21 / 25

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Proof outline for main result

dH((Xi, i)) ≥ t3 − 20t δ(H) ≥ t3 − 20t According to the theorem of Daykin and Häggkvist; H has t independent edges provided δ(H) > 3

4(t3 − 1).

Theorem 6 Let 0 ≤ d < k and let H be an r-partite r-uniform hypergraph with minimum degree δ(H) and |V(H)| = rk. If δ(H) > r − 1 r

  • kr−1 − (k − d)r−1

, then H has more than d independent edges.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 21 / 25

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Proof outline for main result

δ(H) ≥ t3 − 20t

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 22 / 25

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Proof outline for main result

δ(H) ≥ t3 − 20t > 3 4(t3 − 1)

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 22 / 25

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Proof outline for main result

δ(H) ≥ t3 − 20t > 3 4(t3 − 1) Let {e1, . . . , et} be t independent edges in H where ei = ((Xi, i), b, c, d).

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 22 / 25

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Proof outline for main result

δ(H) ≥ t3 − 20t > 3 4(t3 − 1) Let {e1, . . . , et} be t independent edges in H where ei = ((Xi, i), b, c, d). Set Si = {i, b, c, d} for each i.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 22 / 25

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SLIDE 65

Completing partial latin boxes

Current Section

1

Introduction

2

Strictly Completing

3

Main Result

4

Proof outline for main result

5

Completing partial latin boxes

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 23 / 25

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SLIDE 66

Completing partial latin boxes

Theorem 7 Let t ≥ 9. Let P be a 2 × 4t × 4t partial latin box with at most 2t − 1

  • entries. Then P can be completed.

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 24 / 25

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Completing partial latin boxes

THANK YOU FOR YOUR ATTENTION!

Jaromy Kuhl (UWF) Strictly Completing Partial Latin Squares May 2011 25 / 25