We give axiomatic foundations for infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. Continuing work of Higgs and Oxley, this completes the solution to a problem of Rado of 1966. (C) 2013 Henning Bruhn, Reinhard Diestel, Matthias Kriesell, Rudi Pendavingh and Paul Wollan. Published by Elsevier Inc. All rights reserved

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4,5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.

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The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a 'singular' approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassen's results about 'finitary' duality for infinite graphs to full duality, including his extensions of Whitney's theorem.

We prove χ s (G) 1.93Δ(G) 2 for graphs of sufficiently large maximum degree where χ s (G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.