Linear algebra version of Haj´ os 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´ os Theorem
Plan for the talk Introduction History of Haj´ os Theorem General Homomorphism Concept Examples Complexity? Characterization Theorems Tommy R. Jensen Linear algebra version of Haj´ os 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´ os 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´ os constructively characterized the set H ( k ) of graphs that are not colorable with fewer than k colors: The complete graph K k 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 G 1 , G 2 ∈ H ( k ) , and H is obtained from G 1 and G 2 by Haj´ os Construction, then H ∈ H ( k ) . Tommy R. Jensen Linear algebra version of Haj´ os 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´ os constructively characterized the set H ( k ) of graphs that are not colorable with fewer than k colors: The complete graph K k 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 G 1 , G 2 ∈ H ( k ) , and H is obtained from G 1 and G 2 by Haj´ os Construction, then H ∈ H ( k ) . Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Introduction Haj´ os Construction applied to two copies of K 4 This construction was first studied by Dirac in 1957; it is sometimes called the Dirac-Haj´ os Construction. Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Introduction Haj´ os Construction applied to two copies of K 4 This construction was first studied by Dirac in 1957; it is sometimes called the Dirac-Haj´ os Construction. Tommy R. Jensen Linear algebra version of Haj´ os 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 K k 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´ os Theorem
History The Haj´ os 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´ os Theorem
History The Haj´ os 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´ os Theorem
History Ore introduced a more restrictive variation of the Haj´ os 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 of intermediate graphs. From this version he could deduce Brooks’ Theorem. Tutte 1992 Tommy R. Jensen Linear algebra version of Haj´ os Theorem
History Ore introduced a more restrictive variation of the Haj´ os 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 of intermediate graphs. From this version he could deduce Brooks’ Theorem. Tutte 1992 Tommy R. Jensen Linear algebra version of Haj´ os 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´ os Theorem
History Pitassi and Urquhart replaced the Haj´ os 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´ os Theorem
History There exists a construction, that uses elimination, of all 4-chromatic planar graphs starting from K 4 , 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´ os Theorem
History There are versions of the Haj´ os Theorem for list coloring. Gravier 1996, Kr´ al 2004 And for circular coloring. Zhu 2001, 2003 Tommy R. Jensen Linear algebra version of Haj´ os Theorem
History There are versions of the Haj´ os Theorem for list coloring. Gravier 1996, Kr´ al 2004 And for circular coloring. Zhu 2001, 2003 Tommy R. Jensen Linear algebra version of Haj´ os Theorem
History There is a version of Haj´ os Theorem for matroids representable over finite fields. Jaeger 1981 Tommy R. Jensen Linear algebra version of Haj´ os Theorem
We study a generalization of graph coloring and prove characterization theorems that are more general versions of Haj´ os’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´ os Theorem
A General Homomorphism Concept Example Is there a linear function R 2 → R 2 that maps all three colored lines to the red line? Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Definitions – subspaces Tommy R. Jensen Linear algebra version of Haj´ os 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 ). dim V � Gr ( V ) = Gr ( d , V ) is the set of all subspaces. d = 0 Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Hermann G. Grassmann 1809–77 Tommy R. Jensen Linear algebra version of Haj´ os 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 ). dim V � Gr ( V ) = Gr ( d , V ) is the set of all subspaces. d = 0 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. � x 1 , x 2 , . . . , x n � is the subspace spanned by elements x 1 , x 2 , . . . , x n of V . Tommy R. Jensen Linear algebra version of Haj´ os 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´ os Theorem
Examples We can translate certain classical graph and hypergraph problems into this language. Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Example 1: Graph Coloring Definition Let G = ( V G , E G ) be a finite graph, and let k ∈ N . A k -coloring of G is a function c : V G → C , where C is any set of 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 V G is a basis for a vector space V . For each edge e = uv ∈ E G let ℓ e = � u − v � , and let S = { ℓ e : e ∈ E G } . The following equivalence holds. χ ( G ) ≤ k ⇔ S → F . (1) Tommy R. Jensen Linear algebra version of Haj´ os Theorem
Example 1: Graph Coloring Definition Let G = ( V G , E G ) be a finite graph, and let k ∈ N . A k -coloring of G is a function c : V G → C , where C is any set of 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 V G is a basis for a vector space V . For each edge e = uv ∈ E G let ℓ e = � u − v � , and let S = { ℓ e : e ∈ E G } . The following equivalence holds. χ ( G ) ≤ k ⇔ S → F . (1) Tommy R. Jensen Linear algebra version of Haj´ os Theorem
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