3,354 research outputs found
End spaces of graphs are normal
We show that the topological space of any infinite graph and its ends is
normal. In particular, end spaces themselves are normal.Comment: 8 page
Cycle decompositions: from graphs to continua
We generalise a fundamental graph-theoretical fact, stating that every
element of the cycle space of a graph is a sum of edge-disjoint cycles, to
arbitrary continua. To achieve this we replace graph cycles by topological
circles, and replace the cycle space of a graph by a new homology group for
continua which is a quotient of the first singular homology group . This
homology seems to be particularly apt for studying spaces with infinitely
generated , e.g. infinite graphs or fractals.Comment: Advances in Mathematics (2011
Normality of cut polytopes of graphs is a minor closed property
Sturmfels-Sullivant conjectured that the cut polytope of a graph is normal if
and only if the graph has no K_5 minor. In the present paper, it is proved that
the normality of cut polytopes of graphs is a minor closed property. By using
this result, we have large classes of normal cut polytopes. Moreover, it turns
out that, in order to study the conjecture, it is enough to consider
4-connected plane triangulations.Comment: 11 pages, References added, notation improve
Infinite matroids in graphs
It has recently been shown that infinite matroids can be axiomatized in a way
that is very similar to finite matroids and permits duality. This was
previously thought impossible, since finitary infinite matroids must have
non-finitary duals. In this paper we illustrate the new theory by exhibiting
its implications for the cycle and bond matroids of infinite graphs. We also
describe their algebraic cycle matroids, those whose circuits are the finite
cycles and double rays, and determine their duals. Finally, we give a
sufficient condition for a matroid to be representable in a sense adapted to
infinite matroids. Which graphic matroids are representable in this sense
remains an open question.Comment: Figure correcte
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