Linear algebra version of Haj os Theorem Tommy R. Jensen - - PowerPoint PPT Presentation

Linear algebra version of Haj´

• s Theorem

Tommy R. Jensen

Department of Mathematics Kyungpook National University

• Rep. of Korea

GT2015 August 27, 2015 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Plan for the talk Introduction History of Haj´

• s Theorem

General Homomorphism Concept Examples Complexity? Characterization Theorems Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Introduction

Graph Coloring is a classical area of Graph Theory. A 3-coloring of the Petersen graph. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Introduction

For each number k > 2 it is an NP-complete problem to decide whether a given graph is k-colorable. In 1961 Haj´

• s constructively characterized the set H(k) of

graphs that are not colorable with fewer than k colors: The complete graph Kk is in H(k), if G is in H(k), and H is obtained from G by identification of two non-adjacent vertices, then H is in H(k), and if G1, G2 ∈ H(k), and H is obtained from G1 and G2 by Haj´

• s Construction, then H ∈ H(k).

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Introduction

For each number k > 2 it is an NP-complete problem to decide whether a given graph is k-colorable. In 1961 Haj´

• s constructively characterized the set H(k) of

graphs that are not colorable with fewer than k colors: The complete graph Kk is in H(k), if G is in H(k), and H is obtained from G by identification of two non-adjacent vertices, then H is in H(k), and if G1, G2 ∈ H(k), and H is obtained from G1 and G2 by Haj´

• s Construction, then H ∈ H(k).

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Introduction

Haj´

• s Construction applied to two copies of K4

This construction was first studied by Dirac in 1957; it is sometimes called the Dirac-Haj´

• s Construction.

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Introduction

Haj´

• s Construction applied to two copies of K4

This construction was first studied by Dirac in 1957; it is sometimes called the Dirac-Haj´

• s Construction.

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

Remark on complexity If there is a polynomial P such that every G ∈ H(k) of order n can be constructed from copies of Kk in at most P(n) steps, then the complexity classes NP and co-NP coincide. Mansfield and Welsh 1982 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

The Haj´

• s Construction can be applied to construct planar

4-regular 4-critical graphs, which solves a problem of Dirac and Gallai. Koester 1985 It can be applied to construct a class of triangle-free 4-critical graphs that are hard instances for the 3-colorability decision problem. Liu and Zhang 2006 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

The Haj´

• s Construction can be applied to construct planar

4-regular 4-critical graphs, which solves a problem of Dirac and Gallai. Koester 1985 It can be applied to construct a class of triangle-free 4-critical graphs that are hard instances for the 3-colorability decision problem. Liu and Zhang 2006 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

Ore introduced a more restrictive variation of the Haj´

• s

Construction, with idea to apply it to give proofs of structure theorems for graphs with a given chromatic number. He did not provide any example of such a proof. Ore 1967 Tutte described a variation that involves a condition of criticality

• f intermediate graphs.

From this version he could deduce Brooks’ Theorem. Tutte 1992 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

Ore introduced a more restrictive variation of the Haj´

• s

Construction, with idea to apply it to give proofs of structure theorems for graphs with a given chromatic number. He did not provide any example of such a proof. Ore 1967 Tutte described a variation that involves a condition of criticality

• f intermediate graphs.

From this version he could deduce Brooks’ Theorem. Tutte 1992 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

It is not possible to construct every k-critical graph so that all intermediate graphs remain critical. Hanson, Robinson and Toft 1986 for k ≥ 8,

• J. and Royle 1999 for 4 ≤ k ≤ 7.

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

Pitassi and Urquhart replaced the Haj´

• s construction step by a

more restrictive elimination operation, while allowing addition of new edges and vertices at any step. They showed that the complexity of their construction is polynomially equivalent to the complexity of proving theorems in an extended Frege system of logic. Pitassi and Urquhart 1995 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

There exists a construction, that uses elimination, of all 4-chromatic planar graphs starting from K4, while preserving planarity, so that every intermediate graph of the construction remains planar and 4-chromatic. Iwama, Seto and Tamaki 2010 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

There are versions of the Haj´

• s Theorem for list coloring.

Gravier 1996, Kr´ al 2004 And for circular coloring. Zhu 2001, 2003 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

There are versions of the Haj´

• s Theorem for list coloring.

Gravier 1996, Kr´ al 2004 And for circular coloring. Zhu 2001, 2003 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

History

There is a version of Haj´

• s Theorem for matroids representable
• ver finite fields.

Jaeger 1981 Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

We study a generalization of graph coloring and prove characterization theorems that are more general versions of Haj´

• s’s theorem.

This is part of a joint project with Roberto Corcino’s group at the Cebu Normal University in the Philippines. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

A General Homomorphism Concept

Example Is there a linear function R2 → R2 that maps all three colored lines to the red line? Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Definitions – subspaces

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Definitions – subspaces

F is a field. V is a finite dimensional vector space over F. Gr(d, V) is the d’th Grassmannian — the set of d-dimensional subspaces of V (0 ≤ d ≤ dim V). Gr(V) =

dim V

• d=0

Gr(d, V) is the set of all subspaces. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Hermann G. Grassmann 1809–77

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Definitions – subspaces

F is a field. V is a finite dimensional vector space over F. Gr(d, V) is the d’th Grassmannian — the set of d-dimensional subspaces of V (0 ≤ d ≤ dim V). Gr(V) =

dim V

• d=0

Gr(d, V) is the set of all subspaces. The 0-dimensional subspace of V is trivial. Gr∗(V) is the set Gr(V) \ Gr(0, V) of non-trivial subspaces of V. The elements of Gr(1, V) are lines. x1, x2, . . . , xn is the subspace spanned by elements x1, x2, . . . , xn of V. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Definitions – functions

Let V, W be vector spaces over F. L(V, W) is the set of all linear functions from V to W. If S ⊆ Gr(V) and T ⊆ Gr(W), and there exists ϕ ∈ L(V, W) such that ϕ(S) ⊆ T , then ϕ is a homomorphism from S to T , and S is homomorphic to T . We write S → T if S is homomorphic to T . If T = {T} is a singleton, we write S → T instead of S → T . In particular, S → F means that there exists ϕ ∈ L(V, F) such that ϕ(S) = F (that is, ϕ(S) = {0}) for each S ∈ S. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Examples

We can translate certain classical graph and hypergraph problems into this language. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 1: Graph Coloring

Definition Let G = (VG, EG) be a finite graph, and let k ∈ N. A k-coloring of G is a function c : VG → C, where C is any set

• f size k, such that c(u) = c(v) for each edge uv of G.

The chromatic number χ(G) of G is the least value of k for which a k-coloring of G exists. Assume that F is finite and k = |F|. We may assume that VG is a basis for a vector space V. For each edge e = uv ∈ EG let ℓe = u − v, and let S = {ℓe : e ∈ EG}. The following equivalence holds. χ(G) ≤ k ⇔ S → F. (1) Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 1: Graph Coloring

Definition Let G = (VG, EG) be a finite graph, and let k ∈ N. A k-coloring of G is a function c : VG → C, where C is any set

• f size k, such that c(u) = c(v) for each edge uv of G.

The chromatic number χ(G) of G is the least value of k for which a k-coloring of G exists. Assume that F is finite and k = |F|. We may assume that VG is a basis for a vector space V. For each edge e = uv ∈ EG let ℓe = u − v, and let S = {ℓe : e ∈ EG}. The following equivalence holds. χ(G) ≤ k ⇔ S → F. (1) Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 1: Graph Coloring

Proof of (1) For ⇒: assume the existence of a k-coloring c : VG → F of G. Let ϕ : V → F be the linear function given by ϕ(v) = c(v) for each v ∈ VG. Then for each e = uv ∈ EG, ϕ(u − v) = ϕ(u) − ϕ(v) = c(u) − c(v) = 0, since c(u) = c(v). Hence ϕ(ℓe) = F, for each ℓe ∈ S, and S → F follows. The converse ⇐ is similar. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 2: Hypergraph Coloring

Definition H = (VH, EH) be a nonempty finite hypergraph, that is, VH is a finite set, and each element of the set of hyperedges EH is a nonempty subset of VH. For natural number k, a k-coloring of H is a function c : VH → C, where C is a set of size k, such that for every hyperedge h ∈ EH its image c(h) contains at least two distinct colors. The chromatic number χ(H) of H is the least number k for which a k-coloring exists. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 2: Hypergraph Coloring

A 2-coloring of a hypergraph. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 2: Hypergraph Coloring

Assume that VH is a basis for V. For each hyperedge h ∈ EH let Sh be the subspace of V defined by Sh =

• v∈h

αvv :

• v∈h

αv = 0

• .

Let S = {Sh : h ∈ EH}. Then χ(H) ≤ k ⇔ S → F. (2) Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 3: Nowhere-Zero Flow

Definition An orientation G of G is an assignment to each edge e = uv of G of a head h(e) and a tail t(e), that is, G(e) = (h(e), t(e)), where {h(e), t(e)} = {u, v}. A nowhere-zero k-flow in G is a pair ( G, ψ), where G is an

• rientation of G and ψ : EG → {−k + 1, . . . , −1, 1, . . . , k − 1}

satisfies Kirchhoff’s Law of flow conservation:

• {e∈E : h(e)=v}

ψ(e) −

• {e∈E : t(e)=v}

ψ(e) = 0 for all v ∈ VG. If A is any additive Abelian group, a nowhere-zero A-flow in G is defined similarly, that is, ψ is a function from EG to A \ {0A}, satisfying flow conservation at every vertex, where 0A is the zero-element of A. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 3: Nowhere-Zero Flow

Tutte (1953) proved that there exists a nowhere-zero k-flow in G if and only if there exists a nowhere-zero A-flow in G for every Abelian group A with precisely k elements. Tutte also proved that ‘every Abelian group’ can be replaced by ‘any Abelian group’. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 3: Nowhere-Zero Flow

Assume that EG is a basis for the vector space W over F. Then there is a subspace F of L(W, F) that contains precisely those functions that satisfy the Kirchhoff condition. To each edge e of G there is an element ge ∈ L(F, F) defined by ge(f) = f(e), for all f ∈ F. Let ℓe = ge. Then {ℓe : e ∈ EG} → F ⇔ G admits a nowhere-zero k-flow. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 4: Zero-Sum Flow

Definition The undirected analogue of a nowhere-zero A-flow in a graph G, for an Abelian group A, is called a zero-sum A-flow. It is a function ψ : EG → A \ {0A} with the property

• uv∈EG

ψ(uv) = 0 for all v ∈ VG, that is, the sum of the values of ψ on the edges incident to any vertex v of G is zero. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Example 4: Zero-Sum Flow

Let Z be the set of functions ψ : EG → F for which

• {u∈VG : uv∈EG}

ψ(uv) = 0 for all v ∈ VG. For each e ∈ EG define ze : Z → F by ze(ψ) = ψ(e), and let R = {ze : e ∈ EG}. Then G admits a zero-sum F-flow ⇔ R → F. (3) Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Complexity

Proposition 4 Fix a positive integer d and let k = |F|. The problem of deciding S → F for input n ∈ N and S ⊆ Gr(d, Fn) is in P if (d, k) = (1, 2), and

• therwise it is NP-complete.

Proof of Proposition 4 The NP-completeness follows from Examples 1 and 2, since Graph k-Colorability is NP-complete for k > 2, and Hypergraph k-Colorability is NP-complete for k > 1. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Complexity

Proposition 4 Fix a positive integer d and let k = |F|. The problem of deciding S → F for input n ∈ N and S ⊆ Gr(d, Fn) is in P if (d, k) = (1, 2), and

• therwise it is NP-complete.

Proof of Proposition 4 The NP-completeness follows from Examples 1 and 2, since Graph k-Colorability is NP-complete for k > 2, and Hypergraph k-Colorability is NP-complete for k > 1. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Complexity

Proof of Proposition 4 In the case (d, k) = (1, 2), let S ⊆ Gr(1, Fn), where |F| = 2. Assume that S contains the lines v1, v2, . . . , vr, where v1, v2, . . . , vr ∈ Fn \ {0}. Let X ⊆ Fn be the subspace X = v1, v2, . . . , vr. It is easy to calculate a basis B ⊆ {v1, v2, . . . , vr} for X. Then each vi is a sum of the elements of an easily calculated subset Ai ⊆ B, for 1 ≤ i ≤ r. Any function ϕ ∈ L(Fn, F) that satisfies ϕ(vi) = F satisfies ϕ(vi) = 1, so by linearity of ϕ, and since the characteristic of F is 2, S → F holds if and only if Ai has odd size for each i = 1, 2, . . . , r and this is easily checked in polynomial time. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

For simplicity we focus only on the set Gr(1, V) of lines in V. Elimination Assume P, Q ⊆ Gr(1, V) with P \ Q = {ℓ1} and Q \ P = {ℓ2}. Assume there exist x1, x2 ∈ V such that ℓ1 = x1 and ℓ2 = x2, and the line ℓ = x1 + x2 is an element of P ∩ Q. Then R = P ∩ Q is obtained from (P, Q) by elimination. Proposition 5 If R is obtained from (P, Q) by elimination, then R → F ⇒ P → F ∨ Q → F. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

For simplicity we focus only on the set Gr(1, V) of lines in V. Elimination Assume P, Q ⊆ Gr(1, V) with P \ Q = {ℓ1} and Q \ P = {ℓ2}. Assume there exist x1, x2 ∈ V such that ℓ1 = x1 and ℓ2 = x2, and the line ℓ = x1 + x2 is an element of P ∩ Q. Then R = P ∩ Q is obtained from (P, Q) by elimination. Proposition 5 If R is obtained from (P, Q) by elimination, then R → F ⇒ P → F ∨ Q → F. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

For simplicity we focus only on the set Gr(1, V) of lines in V. Elimination Assume P, Q ⊆ Gr(1, V) with P \ Q = {ℓ1} and Q \ P = {ℓ2}. Assume there exist x1, x2 ∈ V such that ℓ1 = x1 and ℓ2 = x2, and the line ℓ = x1 + x2 is an element of P ∩ Q. Then R = P ∩ Q is obtained from (P, Q) by elimination. Proposition 5 If R is obtained from (P, Q) by elimination, then R → F ⇒ P → F ∨ Q → F. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Co-pencils Let S be a subspace of V. The co-pencil of S in V is the set CPS = {ℓ ∈ Gr(1, V) : ℓ ⊆ S}

• f lines not contained in S.

The co-pencil CPS is strong if dim V ≥ 2 + dim S. Proposition 6 If CPS is a strong co-pencil, and U is any 2-dimensional vector space, then Gr(1, U) → CPS. Proposition 7 If S is any subspace of V, then CPS → F, or CPS is strong; not bothTM. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Co-pencils Let S be a subspace of V. The co-pencil of S in V is the set CPS = {ℓ ∈ Gr(1, V) : ℓ ⊆ S}

• f lines not contained in S.

The co-pencil CPS is strong if dim V ≥ 2 + dim S. Proposition 6 If CPS is a strong co-pencil, and U is any 2-dimensional vector space, then Gr(1, U) → CPS. Proposition 7 If S is any subspace of V, then CPS → F, or CPS is strong; not bothTM. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Co-pencils Let S be a subspace of V. The co-pencil of S in V is the set CPS = {ℓ ∈ Gr(1, V) : ℓ ⊆ S}

• f lines not contained in S.

The co-pencil CPS is strong if dim V ≥ 2 + dim S. Proposition 6 If CPS is a strong co-pencil, and U is any 2-dimensional vector space, then Gr(1, U) → CPS. Proposition 7 If S is any subspace of V, then CPS → F, or CPS is strong; not bothTM. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let F be a finite field and let V be a finite dimensional vector space over F. Let P ⊆ Gr(1, V). Then P is G1-constructible if there exist a natural number N ≥ 1 and a sequence R1, R2, . . . , RN = P, such that for all i = 1, 2, . . . , N, (i) Ri is a strong co-pencil, or (ii) Ri is obtained from (Rj1, Rj2) by elimination, for some j1, j2 with 1 ≤ j1, j2 < i. Theorem 1 Either P → F, or P is G1-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let F be a finite field and let V be a finite dimensional vector space over F. Let P ⊆ Gr(1, V). Then P is G1-constructible if there exist a natural number N ≥ 1 and a sequence R1, R2, . . . , RN = P, such that for all i = 1, 2, . . . , N, (i) Ri is a strong co-pencil, or (ii) Ri is obtained from (Rj1, Rj2) by elimination, for some j1, j2 with 1 ≤ j1, j2 < i. Theorem 1 Either P → F, or P is G1-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Proof of Theorem 1, “not both” The “not both” follows inductively from Propositions 5 and 7. Proof of Theorem 1, “either or” Assume P → F. Let F be the set of all elements v of V for which v ∈ P. Clearly 0 ∈ F, so F is nonempty, and if v ∈ F, then v ⊆ F. The proof of “only if” proceeds by induction on n = |F| ≥ 1. Case 1: F is a subspace of V. Then P = CPF, and we let N = 1 and R1 = P. Then R1 is the co-pencil of F, and by Proposition 7, R1 is strong. Hence the statement of the theorem is satisfied. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Proof of Theorem 1, “not both” The “not both” follows inductively from Propositions 5 and 7. Proof of Theorem 1, “either or” Assume P → F. Let F be the set of all elements v of V for which v ∈ P. Clearly 0 ∈ F, so F is nonempty, and if v ∈ F, then v ⊆ F. The proof of “only if” proceeds by induction on n = |F| ≥ 1. Case 1: F is a subspace of V. Then P = CPF, and we let N = 1 and R1 = P. Then R1 is the co-pencil of F, and by Proposition 7, R1 is strong. Hence the statement of the theorem is satisfied. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Proof of Theorem 1, “either or” Case 2: F is not a vector space. Choose f1, f2 ∈ F so that f1 + f2 ∈ F. By the induction hypothesis applied to P ∪ {f1}, there exist a natural number N1 and a sequence R1, R2, . . . , RN1, with RN1 = P ∪ {f1}, such that (i) and (ii) hold for all i = 1, 2, . . . , N1. Similarly, there exist a natural number N2 and a sequence RN1+1, RN1+2, . . . , RN1+N2, with RN1+N2 = P ∪ {f2}, such that (i) and (ii) hold for i = N1 + 1, N1 + 2, . . . , N1 + N2. Since f1 + f2 ∈ F implies f1 + f2 ∈ P, we obtain P from (P ∪ {f1}, P ∪ {f2}) by elimination. Hence with N = N1 + N2 + 1 and RN = P, the sequence R1, R2, . . . , RN satisfies the statement of the theorem.

• Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definintion Let F be a finite field and let V be a vector space over F of finite dimension. Let P ⊆ Gr(1, V). Then P is G2-constructible if there is a natural number N ≥ 0 and a sequence R0, R1, . . . , RN = P, for which R0 = Gr(1, F2), and such that for all i = 1, 2, . . . , N, (i) Rj → Ri for some j with 0 ≤ j < i, (ii) Ri is obtained from (Rj1, Rj2) by elimination, for some j1, j2 with 1 ≤ j1, j2 < i. Theorem 2 Either P → F, or P is G2-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definintion Let F be a finite field and let V be a vector space over F of finite dimension. Let P ⊆ Gr(1, V). Then P is G2-constructible if there is a natural number N ≥ 0 and a sequence R0, R1, . . . , RN = P, for which R0 = Gr(1, F2), and such that for all i = 1, 2, . . . , N, (i) Rj → Ri for some j with 0 ≤ j < i, (ii) Ri is obtained from (Rj1, Rj2) by elimination, for some j1, j2 with 1 ≤ j1, j2 < i. Theorem 2 Either P → F, or P is G2-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Remark Theorem 2 generalizes a version of the Haj´

• s Theorem

presented by Pitassi and Urquhart in 1992. They proved that their construction has the same complexity as the proof complexity of ’Extended Frege Proof Systems’ for proving theorems in mathematics. Proof of Theorem 2 The “not both” follows from Propositions 1–3 and 5. To show “either or”, let R1, . . . , RN be any sequence as in Theorem 1, and let R0 = Gr(1, F2). It follows from Theorem 1 and Proposition 6 that R0, R1, . . . , RN satisfy (i) and (ii).

• Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Remark Theorem 2 generalizes a version of the Haj´

• s Theorem

presented by Pitassi and Urquhart in 1992. They proved that their construction has the same complexity as the proof complexity of ’Extended Frege Proof Systems’ for proving theorems in mathematics. Proof of Theorem 2 The “not both” follows from Propositions 1–3 and 5. To show “either or”, let R1, . . . , RN be any sequence as in Theorem 1, and let R0 = Gr(1, F2). It follows from Theorem 1 and Proposition 6 that R0, R1, . . . , RN satisfy (i) and (ii).

• Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let S1, S2 ⊆ Gr(1, V) be two sets of lines in V. A set of lines S ⊆ Gr(1, V) is a 2-sum of S1 and S2 if there exist Vi, ℓi, xi for i = 1, 2 for which V1, V2 are subspaces of V such that V = V1 ⊕ V2, Si ⊆ Gr(1, Vi), for i = 1, 2, ℓi ∈ Si, for i = 1, 2, ℓi = xi, for i = 1, 2, and S = (S1 ∪ S2 ∪ {ℓ}) \ {ℓ1, ℓ2}, where ℓ = x1 + x2. Proposition 8 If S is a 2-sum of S1 and S2, and if Si → F, for i = 1, 2, then S → F. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let S1, S2 ⊆ Gr(1, V) be two sets of lines in V. A set of lines S ⊆ Gr(1, V) is a 2-sum of S1 and S2 if there exist Vi, ℓi, xi for i = 1, 2 for which V1, V2 are subspaces of V such that V = V1 ⊕ V2, Si ⊆ Gr(1, Vi), for i = 1, 2, ℓi ∈ Si, for i = 1, 2, ℓi = xi, for i = 1, 2, and S = (S1 ∪ S2 ∪ {ℓ}) \ {ℓ1, ℓ2}, where ℓ = x1 + x2. Proposition 8 If S is a 2-sum of S1 and S2, and if Si → F, for i = 1, 2, then S → F. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Proposition 9 If R is obtained from (P, Q) by elimination, then there exists a sequence S1, S2, . . . , S6 with S1 = P, S2 = Q, and S6 = R, such that for i = 3, 4, 5, 6, (i) Sj → Si for some j with 1 ≤ j < i, or (ii) Si is a 2-sum of Sj1 and Sj2 for some j1, j2 with 1 ≤ j1, j2 < i. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let F be a finite field and let V be a vector space over F of finite dimension. Let P ⊆ Gr(1, V). Then P is G3-constructible if there is a natural number N ≥ 0 and a sequence R0, R1, . . . , RN = P, for which R0 = Gr(1, F2), such that for all i = 1, 2, . . . , N, (i) Rj → Ri for some j with 0 ≤ j < i, (ii) Ri is a 2-sum of Rj1 and Rj2 for some j1, j2 with 0 ≤ j1, j2 < i. Theorem 3 Either P → F, or P is G3-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Characterization Theorems

Definition Let F be a finite field and let V be a vector space over F of finite dimension. Let P ⊆ Gr(1, V). Then P is G3-constructible if there is a natural number N ≥ 0 and a sequence R0, R1, . . . , RN = P, for which R0 = Gr(1, F2), such that for all i = 1, 2, . . . , N, (i) Rj → Ri for some j with 0 ≤ j < i, (ii) Ri is a 2-sum of Rj1 and Rj2 for some j1, j2 with 0 ≤ j1, j2 < i. Theorem 3 Either P → F, or P is G3-constructible; not both. Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Remark Theorem 3 is a generalized version of the original Haj´

• s

Theorem from 1961. The → corresponds to vertex identification, and the 2-sum corresponds to the Haj´

• s Construction step.

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem

Congratulations

• n your retirement

Bjarne!

Relax, have fun; avoid anything that resembles actual work for as long as possible. Enjoy!

Tommy R. Jensen

Linear algebra version of Haj´

• s Theorem