Bicycles and left–right tours in locally finite graphs

Abstract

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

On spanning tree packings of highly edge connected graphs

“…Now let T H be a spanning tree packing of H as claimed in (5) and let A be an a-a ′ -bypass of v in T 1 H ∪ T 2 H . Define a vertex a ′′ ∈ A and an a-a ′′ -subarc A ′ of A as follows.…”

Locally finite graphs with ends: A topological approach, II. Applications

“…Theorem 2.11 (i) implies a positive answer to Problem 2.9 (i) for pedestrian graphs G: the sets from C fin and B fin needed to generate the singleton sets {e} form a thin family [21,Prop. 13], and their sum over all e ∈ E is exactly E.…”

“…Then every singleton set {e} has a decomposition {e} = D + F as in Theorem 2.11 (ii). But unlike for pedestrian graphs, this cannot always be chosen orthogonal: there can be edges e all whose decompositions {e} = D + F are such that both D and F (and hence also D ∩ F ) are infinite [21]. Moreover, the family of all the D and F that can be used in such singleton decompositions will not be thin, as soon as G has an infinite bicycle [13].…”

“…Experience has shown that defining these G n as induced subgraphs of G, e.g. those on its first n vertices, is not often a good approach: a badly chosen G n ⊆ G is too oblivious of how it lies inside G. (There might, for example, exist G n -paths 21 in G between vertices of G n that are easily separated in G n .) However, there is a standard approach using finite 'nested' minors rather than subgraphs as G n , which we define now.…”

On the homology of locally compact spaces with ends

“…For the simplicial homology of infinite graphs these standard theorems fail, but this can be remedied: they do work with the cycle space C(G) constructed in [9,10], as amply demonstrated e.g. in [2,3,4,6,5,7,9,8,13,14,17]. This space is built not from finite (elementary) cycles in G itself, as in simplicial homology, but from the (possibly infinite) edge sets of topological circles in the Freudenthal compactification |G| of G, obtained from G by adding its ends.…”