####

#### 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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