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 Kühn (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. We also consider extensions to graphs with infinite degrees.