We consider the equivalence classes of graphs induced by the unsigned
versions of the Reidemeister moves on knot diagrams.
Any graph which is
reducible by some finite sequence of these moves, to a graph with no
edges is called a knot graph. We show that the class of knot graphs
strictly contains the set of delta-wye graphs. We prove that the
dimension of the intersection of the cycle and cocycle spaces is an
effective numerical invariant of these classes

Bollobas, B.

Cooper, C.

Fenner, Trevor

Frieze, A.M.

Journal of Graph Theory

English

Noble, S D

Welsh, D J A

Brunel University Research Archive

A survey of linkless embeddings.

Fourterminal reducibility and projective-planar wye-delta-wye-reducible graphs.

Knots, matroids and the Ising model.

On the computational complexity of the Jones and Tutte polynomials.

On the delta-wye reduction for planar graphs.

Private Communication,

Knot Graphs

Let Gn,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property Ak, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant Ck such that if 2m ≥ Ckn then Ak occurs in Gn,m,k with probability tending to 1 as n → ∞

Birkbeck Institutional Research Online

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least \emphk

Bollobs, B

Cooper, C

Fenner, T I

Frieze, A M

King's Research Portal

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let Gn,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property Ak, if G contains [(k - 1)/2] edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size [n/2]. We prove that, for k ≥ 3, there is a constant Ck such that if 2m ≥ Ckn then Ak occurs in Gn,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 42-59, 2000

Bollobás, B.

Fenner, T. I.

Frieze, A. M.

University of Memphis Digital Commons

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta‐wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

Noble, Steven

Welsh, D.J.A.

Knot graphs

Knot Graphs

Abstract

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Brunel University Research Archive

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King's Research Portal

University of Memphis Digital Commons

Birkbeck Institutional Research Online