Cycle-cocycle partitions and faithful cycle covers for locally finite graphs

By a result of Gallai, every finite graph $G$ has a vertex partition into
two parts each inducing an element of its cycle space. This fails for infinite
graphs if, as usual, the cycle space is defined as the span of the edge
sets of finite cycles in~$G$. However we show that, for the adaptation of
the cycle space to infinite graphs recently introduced by Diestel and
K\"uhn (which involves infinite cycles as well as finite ones), Gallai's theorem
extends to locally finite graphs. Using similar techniques we show that if
Seymour's faithful cycle cover conjecture is true for
finite graphs then it also holds for locally finite graphs
when infinite cyles are allowed in the cover, but not otherwise.

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