Some connections between set theory and computer science

Cowen, Robert

doi:10.1007/bfb0022548

Cited by 10 publications

(10 citation statements)

References 18 publications

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“…This extends a result of R. Cowen [11] who showed that (2, 1)-coloring is NP-complete in general graphs. We show that determining if a planar graph is (3, 1)-colorable is also NP-complete.…”

Section: Hardness Resultssupporting

confidence: 70%

Defective coloring revisited

1997

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A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)-and (5, 1)-colorable, and that a graph of genus γ is (χ γ /(d + 1) + 4, d)-colorable, where χ γ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2, k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k, d)-coloring is NP-complete for k ≥ 3, d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented.

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Section: Hardness Resultssupporting

confidence: 70%

Defective coloring revisited

1997

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“…Furthermore, we establish that it is relatively consistent with ZFA that "for every field F , D(F )" holds, hence "for every field F , SLin(F )" holds, whereas BPI fails. This justifies an unsettled conjecture by Cowen in [4], which states that SLin(Z 2 ) is strictly weaker than BPI.…”

Section: Background Known Results and Aimssupporting

confidence: 82%

“…Theorem 4.7 also settles Cowen's conjecture in [4] that SLin(Z 2 ) is strictly weaker than BPI. We could also derive the result from the forthcoming Theorem 4.8.…”

Section: The Model N 22(p)supporting

confidence: 59%

“…(Hence, B(Q) implies VnD). (4) For every prime natural number p, D(Z p ) implies C(p). (Hence, B(Z p ) implies C(p) for every prime natural number p).…”

Section: Background Known Results and Aimsmentioning

confidence: 99%

“…Cowen also considers the statement SLin(Z 2 ) in his paper [4]. On page 18, he conjectures that SLin(Z 2 ) is strictly weaker than BPI and states that the previous conjecture is still not settled.…”

Section: Existence Of Non-zero Linear Functionals and Systems Of Linementioning

confidence: 99%

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On vector spaces over specific fields without choice

Howard

Tachtsis

2013

Mathematical Logic Qtrly

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We investigate the deductive strength of statements concerning vector spaces over specific fields in the hierarchy of various choice principles. Notation and terminologyAs usual, ω denotes the set of natural numbers, ZF denotes the Zermelo-Fraenkel set theory without AC, and ZFA is ZF with the axiom of extensionality weakened to allow the existence of atoms. In the following, we introduce notation for fragments of the axiom of choice (in some case, the notation goes back to [23]):AC is the axiom of choice.

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Thresholds for Path Colorings of Planar Graphs

Chappell

Gimbel

Hartman

Algorithms and Combinatorics

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Some connections between set theory and computer science

Cited by 10 publications

References 18 publications

Defective coloring revisited

Defective coloring revisited

On vector spaces over specific fields without choice

Thresholds for Path Colorings of Planar Graphs

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