Orthogonality and minimality in the homology of locally finite graphs
The electronic journal of combinatorics, Tome 21 (2014) no. 3
Given a finite set $E$, a subset $D\subseteq E$ (viewed as a function $E\to \mathbb F_2$) is orthogonal to a given subspace $\mathcal F$ of the $\mathbb F_2$-vector space of functions $E\to \mathbb F_2$ as soon as $D$ is orthogonal to every $\subseteq$-minimal element of $\mathcal F$. This fails in general when $E$ is infinite.However, we prove the above statement for the six subspaces $\mathcal F$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).
DOI :
10.37236/3844
Classification :
05C10
Mots-clés : locally finite graph, homology, topological set of edges, orthogonal, cycle space, bond space, cut space
Mots-clés : locally finite graph, homology, topological set of edges, orthogonal, cycle space, bond space, cut space
@article{10_37236_3844,
author = {Reinhard Diestel and Julian Pott},
title = {Orthogonality and minimality in the homology of locally finite graphs},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {3},
doi = {10.37236/3844},
zbl = {1301.05081},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3844/}
}
Reinhard Diestel; Julian Pott. Orthogonality and minimality in the homology of locally finite graphs. The electronic journal of combinatorics, Tome 21 (2014) no. 3. doi: 10.37236/3844
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