Walks through every edge exactly twice II

Chapter the two symmetric subcases that either e lies in some cycle space element, and when we delete e from that, it becomes a cut (this is the statement in (ii)), or else e lies in some cut, and when we delete e from it, this edge set becomes an element of the cycle space (which corresponds to (iii)). If we now naively use the finite version of the cycle space on an infinite, locally finite graph, every element of the cycle space will necessarily be finite, and we will see that this theorem fails for locally finite graphs. One In Chapter 6 we will show that this is also true for locally finite graphs. See also Richter and Shank [22] and Lins, Richter and Shank [19]. In finite graphs, Archdeacon, Bonnington and Little [1] used ladders (which are certain substructures involving left-right tours and bicycles in an unusual way) to give a criterion for planarity. Planarity criteria are very important tools, and this one in particular is purely algebraical and requires

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“…A simple proof of this criterion can be found in Keir and Richter [18]. So, let us extend Theorem 7.1 to locally finite graphs:…”

Section: Next Letmentioning

confidence: 98%

“…In order to define left-right strings, we need to describe what it means to do a 'left' turn followed by a 'right' turn. We will follow the treatment in [18] and [19].…”

Section: Left-right Toursmentioning

confidence: 99%

Bicycles and left–right tours in locally finite graphs

Bruhn

Kosuch

Myint

2009

European Journal of Combinatorics

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“…where ν(I P (C)) denotes the GF (2)-nullity of I P (C) and c(F ) denotes the number of connected components in F . We refer to this equation as the circuitnullity formula or the extended Cohn-Lempel equality; many special cases and reformulations have appeared over the years [5,6,8,15,19,30,32,33,36,37,39,40,43,44,45,50]. A detailed account is given in [46].…”

Section: Thementioning

confidence: 99%

On the linear algebra of local complementation

Traldi

2012

Linear Algebra and its Applications

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“…Also, the version of Cohn and Lempel is restricted to oriented Euler circuits and circuit partitions; the ψ transitions are not relevant to the permutations they considered. Many other authors have rediscovered, refined or restated the circuit-nullity formula in various ways [3,4,9,10,12,14,15,17,22,23,25,26,29,30,31,32,33,34,36].…”

Section: Introductionmentioning

confidence: 99%